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THERMODYNAMICS, 

HEAT   MOTORS, 


REFRIGERATING  MACHINES. 

JOHM  S.  PRELL 
Civil  &  Mechanical  Engineer. 

SAN  FRANCISCO,  CAL. 

BY 

DE  VOLSON  WOOD,  C.E.,  M.A., 

tATE   PROFESSOR   OF  MECHANICAL    ENGINEERING,    STEVENS   INSTITUTE    OF  TECHNOLOGY. 


EIGHTH  EDITION. 
REVISED    AND    ENLARGED. 


NEW  YORK : 

JOHN    WILEY    &   SONS, 

53  EAST  IOTH  STREET. 

1900. 


COFTBIOHTED  BY 

DB  VOLSON  WOOD 


Engineering 
Library 

TJ 


JOHJM  S.  PRELL 

Civil  &  Mechanical  Engineer. 

SAN  FRANCISCO,  CAL. 

PREFACE. 


THE  following  work  has  been  prepared  to  meet  a  want 
experienced  by  myself  in  my  course  of  instruction  in 
Thermodynamics. 

After  reading  several  works  upon  the  subject,  including 
those  of  the  founders  of  the  science— Rankine,  Clausius, 
Thomson — I  was  most  favorably  impressed  with  the  spirit 
of  Rankine's  mode  of  discussing  the  subject.  It  is  in  keep- 
ing with  the  modern  method  of  treating  Analytical  Mechan- 
ics, in  which  the  analysis  is  founded  upon  ideal  conditions 
established  by  definitions,  and  the  resulting  formulas  modi- 
fied to  represent  the  infinite  variety  of  conditions  in  nature. 

But  Rankine's  giant-like  processes  are  not  adapted  to  the 
wants  of  the  average  student.  Article  241  of  his  Steam 
Engine  and  other  Prime  Movers  reaches  the  height  of 
sublimity  in  regard  to  terseness,  comprehensiveness,  and  ob- 
scurity. Without  a  proper  preliminary,  he  crowds  into  a 
few  words  a  principle  which  has  cost  other  writers  protracted 
labor  and  heroic  efforts  to  establish. 

My  aim  has  not  been  to  bring  down  the  subject  to  the 
comprehension  of  the  reader,  but  to  lead  him  up,  by  a  more 
easy  and  uniformly  graded  path,  to  the  same  height,  and  at 
the  same  time  familiarize  him  with  the  way  by  a  free  use  of 
illustrations,  exercises,  historic  references,  and  numerical 
examples. 


733386 


IV  PREFACE. 

The  body  of  the  work  contains  a  development  of  the  es- 
sential principles  of  the  subject,  to  which  I  have  added  an 
Addenda,  for  the  purpose  of  enlarging  upon  the  matter  con- 
tained in  some  of  the  articles,  more  especially  those  pertain- 
ing to  vapors.  This  enabled  me  to  follow  the  thread  of 
the  subject  more  closely  without  turning  aside  to  consider 
applications  to  a  variety  of  substances,  and  to  enlarge  more 
freely  upon  those  secondary  matters  when  separated  from 
the  body  of  the  text. 

Special  attention  is  called  to  the  graphical  representation 
of  internal  work,  as  in  Figs.  36  and  37,  supposed  to  be  new, 
as  well  as  many  of  the  exercises  and  the  discussion  of  En- 
tropy, or  the  Thermodynamic  Function. 

DE  VOLSON  WOOD. 

HOBOKEN,  Sept.,  1888 


THIRD  EDITION. 

THE  treatment  of  the  theoretical  part  of  Thermodynamics, 
including  its  application  to  the  steam  engine,  as  far  as  page 
180,  is  the  same  in  this  edition  as  in  the  first  and  second 
editions.  Since  the  first  edition  there  have  been  added  the 
following  subjects  :  Vapor  Engine  ;  Sterling's  Engine  ; 
Ericsson's  Hot- Air  Engine  ;  Gas  Engine ;  Naphtha  Engine ; 
Ammonia  Engine ;  Steam  Injector ;  Pulsometer  ;  Com- 
pressed-Air  Engine  ;  The  Compressor ;  Steam  Turbine  ;  Re- 
frigerating Machines  ;  Miscellaneous  matter  in  an  Addenda  ; 
Combustion  of  Fuel ;  Steam,  Ammonia  and  other  Tables. 
The  Ammonia  Tables  have  been  computed  from  the  formu- 
las of  the  author  and  are  new. 

THE  AUTHOR. 

August,  1889. 


CONTENTS. 


CHAPTER  I. 

GENERAL    PRINCIPLES. 

ARTICLE  PAG2 

1.  HEAT  is  ENERGY 1 

2.  HEAT  is  NOT  MATERIAL 1 

3.  HEAT  A  RESULT  OP  THE  MOTION  OP  THE  PARTICLES 1 

4.  VELOCITY  OP  HEAT 2 

5.  HEAT  ENERGY  MEASURED  BY  ITS  EFFECTS   2 

6.  THERMAL  UNIT 3 

7.  WORK 3 

8.  INTERNAL  WORK 4 

9.  ACTUAL  ENERGY 4 

10.  LATENT  HEAT 5 

11.  GENERAL  EXPRESSION 5 

12.  TEMPERATURE 5 

13.  THERMOMETERS 6 

14.  THE  AIR-THERMOMETER 7 

15.  A  PERFECT  GAS 8 

16.  AN  ABSOLUTE  SCALE 9 

17.  ABSOLUTE  ZERO  OP  TEMPERATURE 10 

18.  EQUATION  OP  A  GAS 10 

19.  EQUATION  OP  A  PERFECT  GAS 10 

20.  MARIOTTI'S  LAW 12 

21.  LAW  OF  GAY  LUSSAC 12 

22.  SO-CALLED  IMPERFECT  GASES 12 

23.  THERMAL  LINES  . .  .13 


vi  CONTENTS. 

ABTICLE  «" 

24.  ISOTHERMAL  LINES ] 

25.  ADIABATIC  LINES 16 

26.  CYCLE 17 

27.  HEAT  ENGINE • 17 

28.  CARNOT'S  CYCLE 1? 

29.  SOURCE 19 

30.  WORK  DONE 19 

31.  INDICATOR  DIAGRAM 20 

32.  CARNOT'S  CVCLE  is  REVERSIBLE 20 

33.  CONDITIONS  OF  A  REVERSIBLE  CYCLE 21 

34.  HEAT  ABSORBED 22 

35.  MECHANICAL  EQUIVALENT  OF  HEAT 24 

36.  FIRST  LAW  OF  THERMODYNAMICS 27 

37.  THERMAL  CAPACITIES 28 

38.  SPECIFIC  HEAT  AT  CONSTANT  PRESSURE 29 

39.  SPECIFIC  HEAT  AT  CONSTANT  VOLUME 31 

40.  TEMPERATURE  CONSTANT  DURING  EXPANSION 32 

41.  THOMSON  AND  JOULE'S  METHOD 41 

42.  HEAT  ABSORBED  IN  TERMS  OF  EXTERNAL  WORK 43 

43.  HEAT  TRANSMUTED  INTO  WORK 44 

44.  GENERAL  CASE 45 

45.  TEMPERATURE  AND  PRESSURE  AS  INDEPENDENT  VARIABLES.  46 

46.  FUNDAMENTAL  EQUATIONS  OF  THERMODYNAMICS 48 


CHAPTER  II. 

PERFECT    GASES. 

47.  DIFFERENCE  OF  SPECIFIC  HEAT 49 

48.  SPECIFIC  HEAT  CONSTANT 50 

49.  PERFECTNESS  OF  GASES 51 

50.  To  FIND  THE  SPECIFIC  HEAT  OF  A  GAS  AT  CONSTANT  VOLUME.  53 

51.  RELATIVE  SPECIFIC  HEATS 53 

52.  TEMPERATURE  CONSTANT  DURING  EXPANSION 54 

53.  VOLUME  CONSTANT 55 


CONTENTS.  V 

ABTICLB  PAGE 

54.  SIGNIFICATION  OF  R 56 

55.  ARBITRARY  PATH  DURING  EXPANSION 57 

56.  GAS  EXPANDING  WITHOUT  TRANSMISSION  OF  HEAT 61 

57.  ICE  MACHINES 62 

58.  AIR  COMPRESSOR 63 

59.  VELOCITY  OF  SOUND  IN  A  GAS 71 

60.  TO  FIND  THE  VALUE  OF  / 75 

61.  VELOCITY  OF  SOUND  IN  AIR  '. 75 

62.  To  FIND  THE  SPECIFIC  HEAT  OF  A  GAS  BY  MEANS  OF  JOULE'S 

EQUIVALENT 77 

63.  To  FIND  THE  MECHANICAL  EQUIVALENT  OF  HEAT  BY  MEANS 

OF  THE  SPECIFIC  HEAT  OF  A  GAS 78 

64.  RELATIONS  OF  J,  R,  y 79 

65.  OTHER  METHODS  OF  DETERMINING  y 79 

66.  FLOW  OF  GASES 81 

67.  WEIGHT  OF  GAS  DISCHARGED  . .  .83 


CHAPTER  III. 

IMPERFECT     GASES 

68.  GENERAL  DISCUSSION 85 

69.  TEMPERATURE  CONSTANT 8tf 

70.  CHANGE  OF  STATE  OF  AGGREGATION SS 

71.  LATENT  HEAT  OF  FUSION 88 

72.  EXPERIMENTAL  VERIFICATION 91 

73.  EXPANSION  DURING  FUSION 92 

74.  LATENT  HEAT  OF  EVAPORATION 94 

75.  VAPOR 96 

76.  RELATIONS  BETWEEN  THE  TEMPERATURE  AND  PRESSURE  OF 

VAPOR 97 

77.  VOLUME  OF  VAPOR 98 

78.  WEIGHT  OF  VAPOR 99 

79.  EXPERIMENTAL  DETERMINATION  OF  THE  DENSITY  OF  SAT- 

URATED STEAM.  . .                                                                 , .  104 


viii  CONTENTS. 

ARTICLE    V  "« 

80.  MEASUREMENT  OF  HEIGHTS 105 

81.  SUBLIMATION 106 

82.  EVAPORATION  WITHOUT  EBULLITION 106 

83.  SPECIFIC  HEAT  OP  SOLIDS  SENSIBLY  THE  SAME  FOR  PRESS- 

URE CONSTANT  AND  VOLUME  CONSTANT 107 

84.  MECHANICAL  MIXTURES 108 

85.  TOTAL  HEAT  OF  EVAPORATION 110 

86.  EVAPORATIVE  POWER Ill 

87.  SUPERHEATED  STEAM 113 

88.  FREE  EXPANSION 114 

89.  ABSOLUTE  ZERO 116 

90.  GENERAL  EXPRESSION  FOR  SPECIFIC  HEAT 117 

91.  SPECIFIC  HEATS  AT  CHANGE  OF  STATE  OF  AGGREGATION..  117 

92.  MODIFIED  EXPRESSION  FOR  THE  SPECIFIC  HEAT 118 

93.  APPARENT  AND  REAL  SPECIFIC  HEATS, 120 

94.  GENERAL  EXPRESSION  FOR  THE  DIFFERENCE  OF  THE  SPECIFIC 

HEATS 122 

95.  SPECIFIC  HEAT  OF  WATER 126 

96.  ANOTHER  GENERAL  EQUATION  OF  THERMODYNAMICS 126 

97.  OTHER  GENERAL  EQUATIONS 131 

97A.  THE  THERMODYNAMIC  FUNCTION  OR  ENTROPY 136 

98.  LIQUID  AND  ITS  VAPOR 143 

99.  SPECIFIC  HEAT  OF  SATURATED  VAPOR 145 

100.  ADIABATICS  OF  IMPERFECT  GASES 148 

101.  CONDENSERS 154 

102.  ISODIABATIC  LINES . .  157 


CHAPTER  IV. 

HEAT     ENGINES. 

103.  EFFICIENCY 159 

104.  PERFECT  ELEMENTARY  HEAT  ENGINE 159 

1 05.  REGENERATORS  ...  i  an 


CONTENTS.  ix 

ARTICLE  PAGE 

106.  SOME  GENERAL  PRINCIPLES 169 

107.  REMARKS 169 

108.  THE  STEAM-ENGINE—GENERAL  STATEMENT 169 

109.  IDEAL  STEAM  DIAGRAM  171 

110.  STEAM-ENGINE — ISOTHERMAL  EXPANSION 172 

111.  STEAM  ENGINE — ADIABATIC  EXPANSION,  APPROXIMATE  LAW.  175 

112.  ADIABATIC  EXPANSION,  THEORETICAL  LAW 177 

112a.  GENERAL  EQUATIONS  FOR  VAPOR  ENGINES 180 

113.  CUT-OFF 200 

114.  SPECIAL  ENGINES  205 

115.  MULTIPLE  EXPANSIONS 210 

116.  CONDENSATION 212 

117.  EXPERIMENTS  OF  ENGINES 214 

118.  MISCELLANEOUS 217 

119.  HOT- AIR  ENGINES — STIRLING'S  ENGINE 223 

120.  THEORY  OF  STIRLING'S 224 

121.  DESIGNING 230 

122.  ERICSSON'S  HoT-AiR  ENGINE 234 

123.  DESCRIPTION  OF  ERICSSON'S  ENGINE 235 

124.  ANALYSIS 237 

125.  VALUE  OF  THE  RATIO  OF  EXPANSION  TO  PRODUCE  MAXIMUM 

MEAN  EFFECTIVE  PRESSURE 245 

126.  HEAT  ABSORBED  AT  CONSTANT  PRESSURE 246 

127.  GAS  ENGINES 247 

128.  HISTORY 248 

129.  SOME  DETAILS .' 251 

130.  THEORY 253 

181.    FURNACE 256 

132.  WORK  AND  EFFICIENCY 257 

133.  EXPANSION  AND  COMPRESSION  CURVES 258 

134.  EXPERIMENTS 259 

135.  PETROLEUM  ENGINE 266 

136.  EXPERIMENTS 267 

137.  EFFICIENCY  OF  NAPHTHA-ENGINE  PLANT 270 

138.  EFFICIENCY  OF  FLUID  IN  NAPHTHA  ENGINE 271 

139.  REMARKS 273 

140.  AMMONIA  ENGINE . .  274 


x  CONTENTS. 

PAOB 
ARTICLE 

141.  BINARY- VAPOR  ENGINE 

142.  PRODUCTS  OF  COMBUSTION  THE  WORKING  FLUID 278 

143.  THE  STEAM  INJECTOR 2~9 

144.  THEORY  OP  STEAM  INJECTOR & 

145.  APPROXIMATE  FORMULAS  FOR  INJECTOR 288 

146.  INJECTOR  AND  DIRECT-ACTING  PUMP •  289 

147.  THE  PULSOMETER   292 

148.  ANALYSIS  OF  THE  PULSOMETER 293 

149.  COMPRESSED  AIR  ENGINE ". 29(5 

150.  ANALYSIS  OF  COMPRESSED-AIR  ENGINE 296 

151.  AIR  COMPRESSOR 301 

152.  ANALYSIS  OF  THE  COMPRESSOR .  •  •  •  302 

153.  EFFICIENCY •  •  •  305 

154.  FRICTION  OF  AIR  IN  PIPES 306 

155.  STEAM  TURBINE 308 

156.  OUTWARD  FLOW  TURBINE 308 

157.  REACTION  TURBINE 309 

158.  ANALYSIS  OF  OUTWARD  FLOW  TURBINE 314 


CHAPTER  V. 
REFRIGERATION. 

159.  REFRIGERATING  MACHINE 818 

160.  PRACTICAL  OPERATION » 319 

161.  EFFICIENCY 321 

162.  CIRCULATING  FLUID — : 324 

163.  SOME  PROPERTIES  OF  AMMONIA 325 

164.  LATENT  HEAT  OF  EVAPORATION  OF  AMMONIA 328 

165.  SPECIFIC  VOLUME  OF  LIQUID  AMMONIA 833 

166.  SPECIFIC  VOLUME  OF  AMMONIA  GAS 333 

167.  ISOTHERMALS  OF  AMMONIA  VAPOR 333 

168.  ADIABATICS  OF  AMMONIA  VAPOR  834 

169.  SPECIFIC  HEAT  OF  THE  SATURATED  VAPOR  is  NEGATIVE 835 

170.  SPECIFIC  HEAT  OF  LIQUID  AMMONIA 335 

171.  WORK  OF  THE  COMPRESSOR  337 

172.  VOLUME  OF  CYLINDER  FOR  n  POUNDS  OF  VAPOR.  . .  . .  340 


CONTENTS.  XI 

ABTICLE  PAQE 

173.  VOLUME   OF   CYLINDER   FOR    REQUIRED    REFRIGERATING 

EFFECT 340 

174.  DUTY 340 

175.  CASE  OF  SUPERHEATING 344 

176.  EFFICIENCY 347 

177.  TEST  OF  A  COMPRESSOR  SYSTEM , 348 

178.  TEST  OF  AN  ICE-MAKING  PLANT 852 

179.  ABSORPTION  SYSTEM 353 

180.  TEST  OF  ABSORPTION  PLANT 355 

180a.  SULPHUR  DIOXIDE 357 

CHAPTER  VI. 
COMBUSTION. 

181.  ESSENTIAL  PRINCIPLE.  .   358 

182..    HEAT  OF  COMBUSTION 360 

183.  INCOMBUSTIBLE  MATTER 363 

184.  AIR  REQUIRED  FOR  COMBUSTION 364 

185.  FORCED  DRAFT 365 

186.  TEMPERATURE  OF  FIRE. 365 

187.  HEIGHT  OF  CHIMNEY 366 

APPENDIX  I. 
THE  LUMINIFEROUS  ETHER 369 

APPENDIX  II. 
SECOND  LAW  OF  THERMODYNAMICS 389 


ADDENDA. 

MISCELLANEOUS  TOPICS 395 

TABLES 453 

INDEX 472 


THERMODYNAMICS. 

CHAPTEK  I. 

• 

FUNDAMENTAL   PRINCIPLES    AND    GENERAL    EQUATIONS. 

1.  Heat  is  energy.^— Energy  is  a  capacity  for  doing 
work,  and  it  has  been  shown  in  many  ways  that  heat,  by  its 
action  upon  substances,  can  do  work.     Thus,  it   may  cause 
steam  to  drive  a  piston ;  it  causes  solids  and  liquids  to  ex- 
pand, and  changes  the  molecular   condition  of   bodies,  as 
when  solids  are  fused  or  liquids  vaporized.     Heat  is  also 
recognized  as  a  sensation. 

2.  Heat  is  not  material. — A  body  has  the  same 
weight  when  hot  as  when  cold.    Count  Rumford,  in  1T98,  dis- 
covered that  he  could  boil  a  large  quantity  of  water  by  the 
heat  produced  in  boring  a  piece  of  cannon.      Sir  Humphry 
Davy  (about  1799)  melted  ice  by  rubbing  two  pieces  to- 
gether without  heat  being  imparted  to  them. 

3.  Heat  consists  of  a  motion  of  the  particles 
of  a  body. — The  only  known  method  of  directly  meas- 
uring energy  is  by  a  combination  of  mass  and  velocity  ;  thus, 
if  m  be  the  mass  of  a  body  and  v  its  velocity,  then  will  its 
kinetic  energy  be  ^  mv9.     The  mass  being  constant  while  the 
body  is  heated  we  infer  that  its  heat  energy  is  produced  by 
the  velocity  of  its  mass  elements.     These   motions  are  in- 
visible, and  hence  their  character  can  only  be  inferred ;  it 
may  be  a  motion  of  the  ultimate  atom,  or  of  an  atmosphere 


2  THERMODYNAMICS.  [4,  5.] 

about  the  atom,  a  vibratory  or  periodic  motion  of  some  kind, 
or  a  combination  of  simple  motions.  It  is  probably  not  a 
to-and-fro  rectilinear  motion  of  the  molecule.  A  develop- 
ment of  the  theory  of  heat,  fortunately,  does  not  require 
a  knowledge  of  this  motion,  or  even  a  particular  hypothesis, 
beyond  the  fact  that  there  is  a  motion  of  some  kind.  Rankine 
constructed  an  hypothesis  called  "  molecular  vortices,'"  from 
which  he  deduced  many  important  consequences  pertaining 
to  heat.  (See  Edinburgh  Trans.,  vol.  xx.  ;  Philosophical 
Mag.,  1851  and  1855.) 

4.  Velocity  of  heat.— The  perfect   identity   of  the 
laws  of  radiant  heat  with  those   of  light   as  to  reflection, 
refraction,  interference  and  absorption,  and  the  identity  of 
their  velocities,  being  186,300    miles   per  second,  requires 
essentially  the  same  theory  as  to  their  nature  and  mode  of 
propagation.     Electricity  is  also  a  form  of  energy  and  gov- 
erned by  laws  similar  to  those  of  heat.     As  light  is  trans- 
mitted by  means  of  a  subtle  ether  pervading  all  space,  and 
called  the  luminiferous  ether,  so  it  is  believed  that  the  same 
ether  transmits  heat,  electricity  and  magnetism.     (See  Ap- 
pendix.) 

5.  Heat-energy  is  measured  only  by  its  ef- 
fects.— The  kinetic  energy  of  a  mass  may  be  computed  if 
its  mass  and  velocity  are  known,  or  it  may  be  determined  by 
the  work  it  does  in  being  brought  to  rest,  but  since  the  ve- 
locity of  the  particles  producing  heat  cannot  be  measured, 
heat-energy  can  be  measured  only  by  its  effects.     Thus,  if 
a  ball  of  hot  iron  would  just  melt  one  pound  of  ice,  and 
after  being  heated  again  would  just  melt  two  pounds  of  ice, 
then  would  the  ball  in  the  second  case  have  contained  twice 
the  heat  above  the  melting-point  of  ice  that  it  did  in  the 
first  case.     Similarly,  it  requires  about  twice  as  much  heat 
to  raise  the  temperature  of   a  given  amount  of   water  two 
degrees  as  it  does  one  degree.     The  same  principle  applies 


[6,  7.J  THE   THEKMAL   UNIT.  3 

to  other  bodies  of  one  substance  having  different  weights 
and  to  bodies  composed  of  different  substances,  or  to  hetero- 
geneous substances.  For  .scientific  purposes  some  specific 
effect  must  be  assumed  as  a  standard,  and  considered  as  a 
unit. 

6.  The  thermal  unit  is  the  heat  necessary  to  raise  the 
temperature  of  unity  of  weight  of  water  at  its  maximum 
density  one  degree. 

Water  is  at  its  maximum  density  at  39.1°  F.  (4°  C.).  Ac- 
cording to  the  experiments  of  Kopp,  its  volume  is  1.00012 
at  32°  F.,  1.00000  at  39.1°  F.,  1.00011  at  46°  F.,  and  1.04312 
at  212°  F. 

The  British  Thermal  Unit  (B.  T.  U.)  is  the  heat  neces- 
sary to  raise  the  temperature  of  one  pound  of  water  from 
39°  F.  to  40°  F. 

The  French  Calorie  is  the  heat  necessary  to  raise  the  tem- 
perature of  one  kilogramme  of  water  from  4°  C.  to  5°  C.,  and 
is  3.968  times  the  B.  T.  U. 

Some  writers,  in  defining  the  thermal  unit,  start  the  meas- 
urement with  the  temperature  of  melting  ice,  instead  of  at 
39°  F.,  and  although  there  is  but  litile  difference  between 
the  two  values  thus  obtained,  yet  for  scientific  purposes  and 
for  physical  reasons,  the  latter  is  preferable,  and  should  be 
generally  adopted. 

7.  Work. — When  heat-energy  disappears  as  heat,  it  must, 
according  to  the  principles  of  the  conservation  of  energy, 
appear  or  exist  in  some  other  form  of  energy.    When  the  heat 
in  steam  drives  the  piston  of  an  engine,  the  steam  loses  heat 
by    the  operation,  and  an  exact  equivalent  of  the  energy 
so  disappearing   reappears  as   work  done  or   as  energy  in 
the  moving  parts  of  the  engine,  no  allowance  being  made  in 
this  illustration  for  losses  due  to  radiation  or  friction.     To 
aid  one  in  this  conception  conceive  that  one  end  of  the  cyl- 
inder is  filled  with  small,  perfectly  elastic  balls,  bounding 


4  THERMODYNAMICS.  [8,  9.] 

and  rebounding  between  the  head  of  the  cylinder  and  the 
piston ;  they  will,  by  their  continued  action,  produce  a  pres- 
sure upon  the  piston.  If  the  piston  moves  forward  the 
energy  of  the  balls  will  be  diminished,  as  is  shown  in  me- 
chanics in  the  discussion  of  the  impact  of  elastic  bodies,  and 
this  loss  of  energy  will  equal  that  imparted  to  the  piston,  or, 
if  the  piston  moves  uniformly,  equal  to  the  work  done.  In 
general,  when  heat-energy'  disappears  it  is  said  that  an  equiv- 
alent amount  of  work  has  been  done,  although  the  entire 
work  may  not  be  visible  energy.  Some  of  it  may  produce 
molecular  changes  in  the  substance.  In  the  preceding  illus- 
tration, if  the  piston  be  forced  inward  against  the  rebound- 
ing balls,  their  velocity  will  be  increased,  and  hence  their 
energy  will  be  increased  by  an  amount  equal  to  the  work 
imparted  to  them  by  the  piston. 

8.  Internal  work   is  some  kind  of  effect  produced 
upon  the  molecular  character  of  a  substance.     Thus,  if  one 
pound  of  water  at  32°  F.  be  mixed  with  one  Ib.  at  33°  F.  it 
will  produce   two   pounds  of  water  at  32£°  F.,  but  if  one 
pound  of  ice  at  32°  F.  be  mixed  with  1  pound  of  water  at  33° 
F.  the  temperature  of  the  mixture  will  be  32°  F.     Indeed, 
it  is  found  by  experiment  that   it   will    require    about  141 
pounds  of  water  at  33°  to  melt  one  pound  of  ice  producing 
145  pounds  of  water  only  a  very  little  above  32°  F.,  so  that 
nearly  all  the  heat  between  32°  F.  and  33°  F.  in  the   144 
pounds  of  water  is  necessary  to  change  solid  water  (ice)  at 
32°  to  liquid  water  at  the  same  temperature.     Similarly,  a 
large  amount  of  heat  is  absorbed  in  changing  liquid  water 
at  212°  F.  to  gaseous  water  (steam)  at  the  same  temperature. 
This  disgregation  of  the  molecular  structure  is  called  internal 
work,  or  energy  of  a  potential  form. 

9.  The  actual  heat-energy  of  a  substance  is  de- 
pendent  upon   its  temperature.     The  heat  absorbed  by  a 
substance  may  do  external  work,  as  in  driving  a  piston,  and 


[10,  11,  12.]  LATENT   HEAT.  5 

internal  work,  as  shown  in  the  preceding  article,  and  in  ad- 
dition to  both  it  may  increase  the  temperature  of  the  sub- 
stance, thus  increasing  its  energy.  The  last  is  called  actual 
energy.  The  actual  energy  is  some  function  of  the  tempera- 
ture. 

10.  Latent  heat  is  heat  which  produces  effects  other 
than  that  of  change  of  temperature.     Strictly  speaking,  it  is 
not  heat,  but  is  a  measure  of  the  heat  which  has  been  de- 
stroyed in  producing  effects  other  than  that  of  changing  the 
actual  energy  of  the  substance.     Thus,  heat  becomes  latent 
in  producing  changes  in  the  state  of  aggregation  of  the  sub- 
stance, as  infusion,  vaporization  or  sublimation ;  and  as  defined 
in  Article  8,  constitutes  internal  work.     But  it  also  becomes 
latent  in  doing  external  work  by  expansion,  and  if  the  tem- 
perature be  maintained  constant  during  expansion,  the  heat 
destroyed  in   doing  the  work  is  called  the  latent    Jieat  of 
expansion. 

11.  General  expression. — The  total  heat  in  a  defi- 
nite weight  of  any  substance  is  unknown,  although  if  gases 
were   perfect   it   might   be  computed,  as  will  hereafter  be 
shown ;  but  it  is  possible  to  find  expressions  for  the  heat  ab- 
sorbed in  passing  from  one  known  state  to  another,  for  we 
have 

Heat  absorbed  =  change  of  actual  energy  -f-  change  of 

potential  energy  -\-  external  work  / 
=  total  change  of  internal  energy  -\-  exter- 
nal work  j 
=  change  of  actual  energy  -\-  total  work. 

In  this  expression  the  total  internal  energy  includes  all  the 
heat  involved  both  in  changing  the  temperature  and  the  in- 
ternal structure  of  the  substance. 

12.  Temperature  is  a  condition  of  relative  heat.    Ex- 
perience shows  that  when  two  bodies,  one  hotter  than  the 


6  THERMODYNAMICS.  [13.] 

other,  are  near  each  other,  the  hot  body  becomes  cooler  and 
the  cooler  one  hotter.  Heat  of  itself  passes  from  a  hotter  to 
a  colder  body,  and  this  process  cannot  be  reversed  except  by 
an  expenditure  of  mechanical  energy.  The  hotter  body  i« 
said  to  have  a  higher  temperature  than  the  colder  one. 

.  Temperature  is  not  an  indication  of  the  quantity  of  heat 
absorbed  by  a  body,  nor  of  the  amount  of  heat  in  a  body, 
but  of  the  intensity  of  the  heat.  Thus,  if  a  pound  of  iron 
has  the  same  temperature  as  a  pound  of  water,  the  latter  will 
contain  about  eight  times  as  much  heat  as  the  former  for  each 
degree,  as  would  be  found  by  putting  each  pound  into  an- 
other quantity  of  some  liquid  at  a  different  temperature. 

Temperature  is  a  measure  of  the  sensible  heat — that  is, 
actual  heat — which  can  affect  the  senses. 

13.  Thermometers  are  instruments  for  measuring 
differences  of  temperature.  The  more  common  ones  depend 
for  their  action  upon  the  expansibility  of  a  liquid — such  as 
mercury  or  alcohol.  The  liquid  is  confined  in  a  tube  as 
nearly  cylindrical  as  possible,  within  which  it  expands. 
When  the  expansion  of  metals  is  employed  for  determining 
temperature,  the  instruments  used  are  called  pyrometers. 

The  air  thermometer  depends  for  its  action  upon  the 
pressure  produced  by  heat  at  constant  volume. 

All  thermometers  have  two  fixed  points  :  one  the  melting 
point  of  ice,  the  other  the  boiling  point  of  water  at  atmos- 
pheric pressure. 

The  melting  point  of  ice  is  a  more  nearly  fixed  point 
than  the  freezing  point  of  water.  In  some  carefully  con- 
ducted experiments  water  has  been  reduced  several  de- 
grees below  the  ordinary  freezing  point,  32°  F.,  before  freez- 
ing. To  secure  such  a  result,  the  water  must  be  kept  in  a 
condition  of  as  perfect  rest  as  possible.  The  boiling  point 
of  water  depends  upon  the  pressure  to  which  it  is  subjected  ; 
and  since  the  pressure  of  the  atmosphere  is  continually 
changing,  as  shown  by  the  baromoter.  the  pressure  of  one 


[14.]  THE   AIR   THERMOMETER.  7 

atmosphere  must  be  fixed-  for  scientific  purposes.  The  value 
determined  by  Regnault,  and  now  generally  adopted,  is 
2116.2  pounds  per  square  foot,  or  14.7  pounds,  very  nearly, 
per  square  inch.  In  determining  the  boiling  point  of  water  the 
thermometer  should  be  placed  in  the  vapor  near  the  water. 

The  Fahrenheit  scale  has  180  equal  divisions  between  the 
fixed  points,  and  the  zero  of  the  scale  is  32  such  divisions 
below  the  melting  point  of  ice.  It  is  designated  by  F1.,  or 
Fahr. 

The  Centigrade  scale  has  100  equal  divisions  between  the 
fixed  points,  its  zero  being  at  the  lower  or  melting  point  of 
ice.  It  is  indicated  by  C.  It  is  sometimes  called  the  Celsius 
scale. 

The  Reaumur  scale  has  80  equal  divisions  between  the 
fixed  points,  its  zero  being  at  the  lower. 

To  reduce  the  readings  of  one  scale  to  those  of  another, 
the  following  equations  may  be  used  : 

(7=|  (^-32°) ;  F  =  -f  O  -f  32°  ;  R  =  f  C. 

The  construction  here  implied  assumes  that  liquids  ex- 
pand equally  for  equal  quantities  of  heat,  and  that  the 
tubes  containing  them  are  uniform  ;  but  neither  of  these 
conditions  are  exactly  realized,  the  practical  considerations 
of  which  belong  to  Thermometry. 

14.  The  Air  Thermometer. — In  order  to  gain  an 
idea  of  an  elementary  air  thermometer,  conceive  a  small, 
perfectly  cylindrical  tube  closed  at  the  lower  end  to  contain 
a  quantity  of  air,  limited  at  its  upper  end  by  a  drop  of 
mercury  acting  as  a  piston.  Subject  this  instrument  to  the 
temperature  of  melting  ice  under  the  pressure  of  one  atmos- 
phere, 29.922  inches  of  mercury,  and  mark  the  upper  end  of 
the  air  column  ;  then,  next  subject  it  to  the  temperature  of 
boiling  water  under  the  same  pressure  and  mark  the  upper 
end  of  the  air  column  at  this  temperature.  The  two  marks 
will  be  t\\Q  fixed  points  before  described.  If  the  length  of 


8  THERMODYNAMICS.  [15.] 

the  column  from  the  lower  end  to  the  lower  mark  be  unity, 
then  will  its  length  to  the  upper  mark  be  1.3665  as  found  by 
Eegnault.  The  expansion  is  0.3665  of  its  original  volume. 
For  the  Fahrenheit  scale  the  space  between  the  fixed  points 
would  be  divided  into  180  equal  parts,  and  hence  each  part 
would  be  AT3¥ip-  =  0.00203611  of  the  distance  below  the 
lower  fixed  point.  If  the  length  below  the  lower  fixed  point 
be  divided  into  equal  parts  of  the  same  magnitude,  the  num- 
ber of  such  spaces  will  be, 

1  180 


0.00203611        0.3665 


=  491.13. 


If  these  parts  are  numbered  according  to  the  natural 
numbers,  0,  1,  2,  3,  etc.,  beginning  with  zero  at  the  extreme 
lower  end  of  the  tube — called  the  absolute  zero  of  the  air 
thermometer — then  would  the  temperature  of  melting  ice 
be  491.13°  F.  from  the  absolute  zero  of  the  air  thermometer, 
and  that  of  boiling  water  671.13°  F.  from  the  same  zero. 
If  air  were  a  perfect  gas,  this  would  constitute  an  absolute 
scale,  but  as  it  is  not,  a  correction  is  required  in  order  to  es- 
tablish such  a  scale.  For  air  thermometers  the  pressure  at 
constant  volume  is  commonly  used,  instead  of  the  volume  at 
constant  pressure  as  above  described. 

15.  A  perfect  gas  is  defined  to  be  such  that,  under  a 
constant  pressure,  its  rate  of  expansion  would  be  exactly 
equal  to  its  rate  of  increase  of  temperature,  and,  the  volume 
being  constant,  increments  of  pressure  will  be  equal  for 
equal  increments  of  heat.  In  other  words,  it  would  be  a 
substance  in  which  no  internal  work  would  be  done  by 
changes  of  temperature  or  pressure.  No  such  substance  is 
known — it  is  ideal,  subjected  merely  to  a  definition  and  to 
laws  to  be  assigned— and  yet  it  is  of  great  service  in  this 
science.  The  idea  of  a  perfect  gas  was  the  result  of  ex- 
periments upon  existing  gases,  as  air,  oxygen,  hydrogen,  etc., 
which,  at  first,  were  supposed  to  be  represented  by  the  per- 


[16. J  AN    ABSOLUTE   SCALE.  9 

feet  law.  In  mechanics,  at  the  present  time,  the  bodies 
treated  are,  at  first,  the  subjects  of  definition,  and  considered 
perfect,  as  perfect  solids,  perfect  liquids,  perfectly  elastic^ 
etc.,  and  the  results  obtained  from  these  hypotheses  made 
practical  by  the  introduction  of  moduli  the  values  of  which 
are  found  by  experiments.  The  same  method  is  adopted  in 
this  science. 

16.  An  absolute  scale  is  one  whose  divisions  would 
be  indicated  by  a  perfect  gas  thermometer.  On  such  a 
scale  the  divisions  would  be  exactly  equal  for  equal  incre- 
ments of  heat  down  to  the  zero  of  the  scale.  Since  a  per- 
fect gas  is  unknown,  the  zero  of  the  absolute  scale  can  be 
determined  only  approximately  by  computation,  as  will  be 
shown  hereafter,  where  the  best  result  yet  obtained  fixes 
it  at  492. 66°  F.  below  the  melting  point  of  ice.  The  letter 
F.  here  affixed  implies  that  there  are  180  divisions  between 
the  fixed  points,  as  in  Fahrenheit's  scale.  This  zero  on  the 
centigrade  scale  io  -|  of  492.66°  =  273.7°  C.  Temperature 
on  the  absolute  scale  will  generally  be  indicated  by  the 
Greek  letter  r,  and  the  temperature  of  melting  ice  by  r0. 
If  T°  F.  indicate  the  temperature  from  the  zero  of  the  Fah- 
renheit scale,  and  T°  C.  from  the  zero  of  the  centigrade 
scale,  we  will  have 

ra  =  492.66°  F  =  273.7°  C. 
r  -.=  460.66°  F+  T°  F. 
—  273.7°    C+  T  C. 

It  is  found  that  air  is  so  nearly  a  perfect  gas  within  the 
ranges  of  temperature  and  pressure  for  which  it  has  been 
tested  that  it  may  be  considered  as  such  for  all  practical 
purposes,  and  will  be  so  considered  theoretically  except  in 
the  determination  of  the  place  of  the  zero  of  the  absolute 
scale.  Further,  the  ordinary  mercurial  thermometer  agrees 
sufficiently  well  with  the  air  thermometer  for  the  more  or- 
dinary ranges  of  temperature  met  with  in  engineering  prac- 


10  THERMODYNAMICS.  [17,  18,  19.] 

tice  to  be  used  in  such  cases.  But  for  scientific  purposes 
and  for  extreme  cases  in  practice,  the  difference  is  too  large 
to  be  ignored.  Regnault  found  that  when  the  air  ther- 
mometer marked  630°  F.  above  the  melting  point  of  ice,  the 
mercurial  thermometer  indicated  651.9°  above  the  same 
point,  a  difference  of  about  22°.  Liquids  generally  expand 
more  rapidly  the  higher  the  temperature. 

17.  The  absolute  zero  of  temperature  is  the 

zero  of  the  absolute  scale,  and  corresponds  to  the  condition 
of  total  deprivation  of  heat ;  at  which  temperature  no  sub- 
stance could  exercise  any  expansive  power.  This  tempera- 
ture has  never  been  reached,  and  the  nearest  approach  to  it 
has  been  produced  by  expansion  in  liquefying  air,  oxygen  and 
nitrogen,  reaching  — 373°  F.  (—225°  C.),  or  more  than  f  the 
distance  from  the  zero  of  Fahrenheit  to  the  absolute  zero. 
(Complex  Rendus,  Feb.  9,  1865;  Jour.  Frank.  Inst.,  Sept. 
1886,  p.  213).  The  absolute  zero  is  about  492.66—491.13 
=  1.5  degrees  Fahr.  below  the  zero  of  the  air  thermometer, 
as  computed  on  the  hypothesis  of  the  same  rate  of  contrac- 
tion of  air  below  32°  as  from  32°  to  212°.  This  law  might 
change  as  the  temperature  was  extremely  reduced,  but  it 
would  continue  uniform  for  the  ideally  perfect  gas. 

18.  The  equation  of  a  gas  is  an  equation  expressing 
a  continuous  relation  between  its  volume,  pressure  and  tem- 
perature throughout  a  finite  range  of  the  same.     Let  p  be 
the  pressure  on  a  unit  of  area  of  the  substance  when  the 
volume  of  one  pound  is  v  and  absolute  temperature  is  T, 
then,  generally, 

P=f(v,  r\ 

which  may  be  considered  as  the  equation  to  a  surface,  called 
the  therm-odynamic  surface. 

19.  Equation  of  a  perfect  gas According  to  the 

definition  in  Article  15, 

(p\  oo  T,  and  (v)p  oc  r, 


[19.]  EQUATION   OF   A   PERFECT   GAS.  11 

where  the  subscripts  represent  the  quantities  which  are 
constant  while  the  others  vary,  and  combining  these  in  one 
expression,  we  have 

p  v  <x   i, 
or,  pv  _  p1vl  ^  Q 

where  pn  va  rl  are  contemporaneous  fixed  values.  Let  j>0 
be  the  pressure  of  one  atmosphere,  r0  the  absolute  temper- 
ature of  melting  ice,  and  v0  the  volume  of  one  pound  of  the 
gas  at  that  pressure  and  temperature,  then  will  equation  (1) 
become 

£1  =  S>±  =  *(*,),  (2) 

which  is  the  equation  required.  The  values  of  p0  and  Tt 
have  already  been  given  and  are  independent  of  the  nature 
of  the  gas,  but  v0  depends  upon  the  density  of  the  gas.  A 
cubic  foot  of  dry  air  weighs  at  sea  level  at  the  temperature 
of  melting  ice 

y\  =  0.080728  pounds  ; 
hence, 

1 

v,  =  --  =  12.387  ; 
0.080728 

.  p.  va  _     2116.2  X  12.387  _  26214  _  g3  gl 

r0    "  492.66  ~~492^66  ~ 

when  TO  is  measured  from  the  zero  of  the  absolute  scale  ; 
but  if  it  be  from  the  zero  of  the  air  thermometer,  we  have 


^  7 


TO         491.13 
and  equation  (2)  becomes, 

for  the  absolute    scale,  pv  =  53.21  r,  (3) 

for  the  air  thermometer,  p  v  —  53.37  t.  (3r) 

For  French  units,  let 

p*   =  the  pressure  of  one  atmosphere  in  kilogrammes  per  square  metre/ 


12  THERMODYNAMICS.  [20,  21,  22.] 

-BO'  =  the  volume  of  a  kilogramme  of  the  gas  in  cubic  metres, 
TO'  =  the  absolute  temperature  of  melting  ice  on  the  centigrade  scale  ; 
then 

x 


___ 
2.2  (3.28)3       _ 


~r~  fr0  1.8222      r0 

which  for  air  becomes  in  kilogramme—  metre—  centigrade  units  (k.  m.  c.), 
omitting  the  accents, 

pv  =  29.20  T  • 
and  in  decimetre  —  kilo.  —  centigrade  units, 

pv  =  2.920  T. 

20.  Mariotte's   law.  —  If  the   temperature  be  con- 
stant, equation  (2)  shows  that  the  volume  varies  inversely  as 
the  pressure  ;  a  law  discovered  by  direct  experiments  upon 
gases,  and  known  as  Mariotte's  law,  supposed  by  some  to 
have  first  been  discovered  by  that  investigator,  but  by  others 
this  credit  is  given  to  Boyle.     Fora  time  after  the  announce- 
ment of  the  law  it  was  supposed  to  be  perfect  for  the  so- 
called  permanent  gases,  but  more  refined  experiments  have 
shown  that  the  actual  law  governing  them  is  only  a   very 
close  approximation  to  it. 

21.  Law  of  Gay  Lussac  (or  of  Charles).  —  According 
to  equation  (2),  if  the  pressure  be  constant,  the  volume  will 
increase  directly  as  the  temperature,  or 

*•=***, 

P 

a  law  discovered  by  Gay  Lussac  (or,  according  to  some,  by 
Charles)  by  experiments  upon  actual  gases,  and  known  as 
the  law  of  Gay  Lussac.  At  first  it  was  supposed  to  be  the 
perfect  law  of  the  so-called  permanent  gases,  but  it  is  now 
known  not  to  be  exact  though  very  nearly  so.  Originally, 
it  was  not  stated  in  terms  of  the  absolute  temperature,  as  that 
term  was  not  then  known,  but  the  law  of  the  increments  is 
the  same  on  any  thermometric  scale. 

22.  The  so-called  imperfect  gases  include  all 
such  as  cannot  be  represented  with  sufficient  accuracy  by 


[23.]  THERMAL   LINES.  13 

equation  (2).  All  known  gases  are  imperfect,  strictly  speak- 
ing,  but  the  permanent  gases  are  so  nearly  perfect  that  they 
may,  for  engineering  purposes,  be  considered  a«  perfect.  No 
single  formula  can  represent  exactly  the  law  of  imperfect 
gases,  but  the  most  comprehensive  one,  and  one  which  may 
be  made  to  represent  actual  substances  with  sufficient  ac- 
curacy for  practical  purposes,  was  deduced  by  Kankine  from 
his  theory  of  Molecular  Vortices,  and  is 

P^  =  P^_^__  ^__^L_.  <&>.,  (4) 

r          r^          r        r'2       r3 

in  which  #OJ  #„  #„  &c.,  are  functions  of  the  density  to  be 
determined  by  experiment  ;  but  as  the  theory  here  referred  to 
is  not  a  recognized  part  of  science,  the  formula  is  accepted 
only  so  far  as  it  conforms  to  the  results  of  experiment. 
(Kankine's  So.  Papers,  32.) 

For   carbonic  acid  gas  the  form  of  the  equation,  as  con- 
firmed by  the  experiments  of  Regnault,  becomes 

p  v  =$•?•  r  -  J_  =  E  f  -   A,  (5) 

T0  TV  TV 

in  which  p0  =  2116.2,  v,  =  8.15T2,  p0  v0  =  17262,  TO  = 
492.66°,  o  =  481600; 

...^,=  851-^2.  (6) 

Sir  William  Thomson  and  Dr.  Joule  used,  for  imperfect 
gases,  the  formula, 

(7) 


in  which  for  air  the  constants  for  French  units  are 

E  =  2.8659,  a  =  771386,  ft  =  8M560,    y  =  214325840. 

(Phil.  Trans.  [1854],  CXLIT.,  360). 

23.  Thermal  lines.—  Any  line  the  co-ordinates  of 
which  represent  the  contemporaneous  relation  between  the 
pressure,  volume,  and  temperature  of  a  body  subjected  to 


14  THERMODYNAMICS.  [24.] 

thermal  conditions,  is  a  thermal  line.  Ideally,  it  may  be 
any  line  on  a  thermodynamic  surface  ;  actually,  the  projec- 
tion of  a  thermal  line  on  any  one  of  the  co-ordinate  planes 
is  called  a  thermal  line,  and  geometrically  it  is  called  the 
path  of  the  fluid,  although  the  latter  refers  to  the  projec- 
tion on  the  co-ordinate  plane,  pv,  uilless  otherwise  stated. 
Thermal  lines  on  the  plane  pv  constitute  a  diagram  of 
energy.  If  the  pressure  p  be  constant,  the  line  is  called  an 
isobar  •  if  the  volume  v  be  constant,  it  is  called  an  iso- 
metric. Thermal  lines  were  introduced  into  this  science  by 
M.  Clapeyron. 

24:.  Isothermal  lines  represent  the  relation  between 
the  pressure  and  volume  when  the  temperature  is  maintained 
constant.  In  equation  (2)  if  r  be  constant  we  have 

j)v  =  Itr  =  m,  -(8) 

for  the  equation  of  an  isothermal  of  a  perfect  gas.  It  is  an 
equilateral  hyperbola  referred  to  its 
asymptotes  as  shown  in  Fig.  1,  in  which 
O  a  is  the  axis  of  the  hyperbola,  the 
branches  of  which  will  be  asymptotic 
respectively  to  the  axes  0  v  and  Op; 
O  v  being  the  axis  of  volumes  and  Op 
the  axis  of  pressures. 


o 

PIG.  1.  EXERCISES. 

1.   Construct   an   isothermal   for  air 
considered  as  a  perfect  gas. 

Assume  a  temperature  of  60°  F.  or  call  it  520*  P.  absolute,  then 

p  v  =  27669 
v  =  10,        p  =  2766.9 
v  =  100,      p  =  276.69 
v  =  p,         p  =  166.4 
v  =  1000,    p  =  27.669 

&c.  &c. 

These  numbers  are  so  large  we  take  ^n  of  their  values  as  inches,  or  parts 


[24-1  ISOTHERMAL  LINES.  15 

of  an  inch,  and  construct  the  curve  as  in  Fig.  2.     But  for  the  equilateral 
hyperbola  it  is  unnecessary  to  compute  any  co-ordinates  except  for  the  ver- 
tex c  ;  for,  having  found  c  by  making  p  =  v  = 
O  a  =  a  c,  bisect  0  a  at  d  and  make  d  e  =  2  a  c, 
&c.  ;  and  make  0  g  =  2  0  a  and  g  h  =  \ac,  &c. 

2.  Construct  an  isothermal  for  air  at 
the  temperature  of  1°  F.  absolute. 

3.  Find  the  vertex  of  the  hyperbola 
of  the  isothermal  for   air   whose   tem- 
perature is  T  =  400°  F.  O" 

4.  Find  It  for  the  following  gases  :  FIG.  2. 

For  hydrogen,  v0  =  178.83,         R  = 
nitrogen,  va  =  12.75,  R  = 

oxygen,      v0  =  11.20,  R  = 

5.  Find  the  value  of  R  in  French  units  for  hydrogen. 

6.  Find  the  equation  to  the  isothermal  for  carbonic  acid 
gas  for  the  temperature  T  =  60°  F. 

7.  What  is  the  volume  of  air,  considered  as  a  perfect  gas, 
under  the  pressure  of  four  atmospheres  and  absolute  tem- 
perature of  r  =  800°  ? 

8.  If  the  heat  in  one  pound  of  carbon  is  14500  B.  T.  U., 
how  many  pounds  of  carbon  completely  consumed  are  neces- 
sary to  increase  the  temperature  of  2000  pounds  of  water 
45°  F.  ? 

9.  How   many  kilogrammes   of   water  would   be    raised 
25°  C.  by  the  heat  in  one  pound  of  carbon  ? 

10.  On  a  diagram  of  energy  draw  on  the  plane  v  t  the 
locus  of  the  path  of  a  perfect  gas  when  the  pressure  is  con- 
stant. 

11.  Find  the  pressure  per  square  inch  of  two  pounds  oi 
air  when  its  volume  is  one  half  of  a  cubic  metre  and  its  ab- 
solute temperature  is  500°  C. 

12.  Show  that  all  isothermals  of  a  perfect  gas  are  asymp- 
totic to  each  other  as  well  as  to  the  co-ordinate  axes^?  and  v. 

13.  "What  is  the  temperature  of  a  pound  of  air  when  its 


16  THERMODYNAMICS.  L25-) 

volume  is  5  cubic  feet  and  pressure  35  pounds  per  square 
foot? 

14.  What  is  the  weight  of  a  cubic  foot  of  air  when  the  pres- 
sure is  50  pounds  per  inch  and  the  temperature  160°  F  '* 

25.  Adiabatic  or  Tseiitropic  lines  represent  the 
relations  between  the  volume  and  pressure  of  a  substance 
doing  work  by  expansion  without  transmission  of  heat. 
Conceive  a  gaseous  substance  to  be  enclosed  in  a  cylinder 
having  a  frictionless  piston,  it  will,  by  driving  the  piston,  do 
work.  It  will  be  conceived  that  the  external  pressure  is  infin- 
itesimally  less  than  the  internal  during  expansion.  The 
temperature  of  the  enclosed  gaseous  substance  may  depend 
upon  several  conditions.  If  heat  be  properly  supplied  the 
temperature  may  be  maintained  con- 
stant, producing  isothermal  expansion, 
which  may  be  represented  by  the  line 
A  J,  Fig.  3.  Having  performed  that 
operation,  bring  the  substance  to  its 
initial  state  A,  and  conceive  the  ex- 
pansion to  take  place  without  any 
°  o  transmission  of  heat,  to  do  which  the 

vessel  must  be  considered  as  imper- 
meable to  the  passage  of  heat,  in  which  case  the  external 
work  will  be  at  the  expense  of  the  heat-energy  of  the  sub- 
stance, and  therefore  the  temperature  will  fall  as  expansion 
proceeds,  and  the  pressure  will  also  fall  on  account  of  the 
loss  of  temperature,  as  shown  by  equation  (2),  and  the  line 
A  d  representing  the  continuous  relation  between  the  vol- 
ume and  pressure  will  be  lower  than  the  isothermal  A  b,  and 
its  slope  downward  greater  for  equal  volumes.  If  the  sub- 
stance be  compressed  from  state  A,  the  line  A  e  will  be 
above  the  isothermal  1)  A  c.  The  line  e  A  d,  representing 
the  law  of  expansion  or  of  compression  without  transmis- 
sion of  heat,  is  by  Kankine  called  an  Adiabatic  (from 
,  to  pass  through),  and  by  Gibbs,  Clausius  and 


[20,  27,  28.] 


CYCLE. 


17 


others,  Isentropic,  because  the  entropy  (a  term  to  be  con- 
sidered later)  remains  constant  in  this  kind  of  expansion. 
Adiabatics  are  asymptotic  to  the  axes^>  and  v  and  also  to  the 
isothermals. 

26.    If  a  fluid,  after  a  series  of  changes  of  pressures  and 
volumes,  returns  to  its  initial  state,  the  path  of  the  fluid 
will  be  a  re-entrant  curve,  as  A   and 
_Z?,  Fig.  4,  and  in  such  cases  the  fluid  is 
said  to  work  in  a  cycle. 


27.  A  heat  engine  is  a  machine 
for  continuously  transforming  heat  into 

work.     Such  engines  in  practice  work 

-.  FIG.  4. 

in  cycles. 

28.  Carnot's  cycle.      This  is  a  cycle  performed  by 
an  imaginary  heat  engine,  devised  by  M.  Carnot  in  1824:, 
and  involves  the  most  important  fundamental  principle  of 
this  science.     The   following  is  the  operation : 

Let  B,  Fig.  5,  be  a  piston  moving  in  a  frictionless  cylin- 
der, all  parts  of  which  are  perfectly  impermeable  to  the  pas- 
sage of  heat  except  the  base  F.  Let 
the  base  of  the  cylinder  be  one  square 
foot,  so  that  the  height  of  the  piston 
will  correspond  with  the  number  of 
cubic  feet  below  it,  and  let  the  cylinder 
contain  one  pound  of  air,  or  any  other 
gas.  Let  //  be  an  indefinitely  large 
vessel  containing  heat  at  a  given  tem- 
perature, and  L  another  indefinitely  large  vessel  contain- 
ing heat  at  a  lower  temperature,  the  initial  letters,  II 
and  Z,  indicating  the  relative  temperatures.  The  vessels 
are  assumed  to  be  indefinitely  large,  so  that,  in  imparting 
heat  to  a  finite  body,  they  will  maintain  a  sensibly  uniform 
temperature.  Let  N  and  N'  be  plates,  as  large  or  larger 
than  the  base  F  of  the  cylinder,  perfectly  impermeable  to 


FIG.    5. 


18  THERMODYNAMICS.  [28.J 

the  passage  of  heat.  Two  of  these  are  used  simply  for  con- 
venience of  arrangement,  so  that  the  operation  to  be  de- 
scribed, passing  in  the  direction  indicated  by  the  arrows, 
will  be  the  more  suggestive  of  a  cycle.  Conceive  the  base  f 
of  the  cylinder  to  be  placed  against  the  vessel  //  ;  the  pound 
of  air  in  the  cylinder  will  quickly  become  of  the  same  tem- 
perature as  that  of  //.  While  in  this  condition  let  the  pis- 
ton move  outward  against  a  resistance  which  is  continually 
infinitesimally  less  than  the  pressure  within, — the  tempera- 
ture will  be  constantly  that  of  //,  and  the  expansion  will  be 
isothermal. 

After  the  piston  has  been  moved  outward  as  far  as  desired 
in  this  manner,  transfer  the  cylinder  to  the  non-conducting 
cover  2V  and  allow  the  piston  to  move  outward  still  further 
by  a  gradual  reduction  of  the  external  pressure  ; — the  press- 
ure and  temperature  of  the  substance  will  both  fall,  and 
since  the  walls  of  the  cylinder  are  impermeable  to  the  pas- 
sage of  heat,  the  expansion  will  be  adidbatic.  Let  the  op- 
eration be  continued  until  the  temperature  of  the  pound 
of  gas  in  the  cylinder  has  been  reduced  to  that  in  the 
vessel  L, 

At  the  end  of  the  preceding  operation  let  the  cylinder  l>e 
removed  to  the  vessel  Z,  and  the  piston  then  forced  inward ; 
the  heat  generated  by  the  compression  of  the  pound  of  air 
will  escape  as  fast  as  generated,  and  is  said  to  be  rejected  or 
emitted  into  the  vessel  Z,  the  temperature  of  which  will  not 
be  sensibly  changed  ;  hence  the  temperature  of  the  pound  of 
air  will  be  constantly  that  of  the  vessel  Z,  and  the  compres- 
sion will  be  isothermal.  Let  the  operation  continue  to  such 
a  point  that  when  the  cylinder  is  removed  to  the  cover  JV 
and  the  air  compressed  adiabatically  until  the  temperature  is 
raised  to  that  in  the  vessel  //,  the  volume  will  be  the  same 
as  that  at  the  beginning  of  the  series  of  operations. 

To  show  these  operations  graphically,  let  O  I,  Fig.  6,  rep- 
resent the  volume  and  5  B  the  pressure  of  the  pound  of  gas 


[29,  30.  J 


SOURCE. 


19 


FIG.    6. 


in  the  initial  state ;  then  will  B  on  the  diagram  represent 
this  state. 

First  operation.  When  the  cylinder  is  in  contact  with  the 
vessel  77,  the  expansion  of  the  gas  will  be  represented  by  the 
isothermal  B  C,  O  c  being  the  final 
volume  and  c  C  the  final  pressure. 

Second  operation.  When  the  cylin- 
der is  in  contact  with  the  cover  N,  the 
expansion  will  be  represented  by  the 
adiabatic  C  D,  O  d  being  the  final 
volume  and  d  D  the  corresponding 
pressure. 

Third  operation.       The   compres- 
sion, when  the  cylinder  is  in  contact  with  the  vessel  Z,  will 
be  represented  by  the  isothermal  line  D  A. 

Fourth  operation.  The  compression  when  the  cylinder 
is  on  the  cover  N'  will  be  represented  by  the  adiabatic  A  1$. 

These  are  the  successive  operations  as  indicated  by  Car- 
not ;  but  it  is  more  convenient,  in  describing  the  process, 
to  begin  either  at  the  state  C or  A,  on  account  of  limiting 
the  third  operation.  Thus,  when  the  cylinder  is  on  the  ves- 
sel //  and  in  the  state  (7,  let  it  be  transferred  to  N  and  ex- 
panded along  CD  until  the  temperature  is  reduced  to  that 
of  L ;  then  transferred  to  L  and  compressed  along  DA  any 
desired  amount ;  thence  transferred  to  N'  and  compressed 
until  the  temperature  is  raised  to  that  of  II \  then  transferred 
to  II  and  expanded  along  B  C  to  the  state  C. 

29.  Source.  The  vessel  from  which  the  working  sub- 
stance receives  heat,  as  II  in  the  above  operation,  is  called 
the  source.     Similarly,  the  vessel  receiving  the  heat  emitted 
from  the  working  substance,  as  L  in  the  above  operation,  is 
called  the  refrigerator.     In  engineering  science  these  are 
called,  respectively,  the  furnace  and  condense?'. 

30.  Work  done.     During  the  expansion  from  state  B 


20  THERMODYNAMICS.  [31,  32.] 

to  state  C  work  is  done  by  the  gas  while  forcing  the  piston 
outward,  represented  by  the  area  I  B  C'c,  and  while  expand- 
ing from  C  to  D  more  work  is  done  by  the  gas,  represented 
by  the  area  c  CD  d\  but  during  the  compression  from  D  to 
A  work  is  done  by  the  piston  upon  the  gas,  the  amount 
being  represented  by  the  area  d  D  A  a,  and  work  is  still  fur- 
ther done  upon  the  gas  in  compressing  it  from  A  to  B,  rep- 
resented by  the  area  a  A  £  I.  The  difference  between 
these  works  will  be  the  external  work  done  by  the  cycle  of 
operations.  We  have 
+  IB  Cc  +  cC  Dd-  dJ)Aa-aABl  =  A 


31.  Indicator    diagram.     The  diagram  A  B  C  D 
would  be  described  by  an  indicator  on  Garnet's  imaginary 
engine  ;  and  the  area  of  an  actual  indicator  diagram,  taken 
from  any  engine,  expressed  in  foot-pounds,  is  a  measure  of 
the  heat  destroyed  in  the  cycle.     It  is  in  this  sense  that  we 
speak  of  "  foot-pounds  of  heat." 

32.  Cariiot's  cycle  is  reversible.     In  a  complete 
cycle,  if  all  the  heat  taken  in  is  at  one  uniform  tempera- 
ture, and  all  the  heat  rejected  is  at  a  uniform   lower  tem- 
perature, the  operation  is  called   Gamut's   cycle.     Such   a 
cycle  is  reversible,  for  all  the  operations  may  be  performed 
in  precisely  the  reverse  order,  the  final  result,  however,  being 

work  done  by  the  piston  upon  the  gas  in 
the  cylinder,  the  energy  of  the  gas  thereby 
being  increased  by  an  amount  represented 
by  the  area  B  A  1)  C,  Fig.  6,  expressed 
in  foot-pounds.  A  reversible  engine  is  also 
called  a  perfect  engine. 

Non-reversible  cycle.     As  an   example 
of  a   non-reversible  cycle,  after  the   sub- 
stance has  expanded  isothermally  while  in 
communication  with  the   source,    represented  by   the   line 
B  C,  Fig.  7,  let  it  be  transferred  directly  to  the   refriger- 


133.]  CONDITIONS  OF  A  REVERSIBLE   CYCLE.  21 

ator — heat  will  be  abstracted  and  the  pressure  may  be 
reduced  at  constant  volume,  and  hence  without  doing 
work,  the  operation  being  represented  by  the  line  CD. 
Then  compress  it  isothermally  when  in  communication  with 
the  refrigerator  along  the  line  D A ;  then  transfer  it  direct- 
ly to  the  source,  raising  the  temperature  and  pressure  to  the 
initial  state  B.  This  cycle  cannot  be  performed  in  precisely 
the  reverse  order ;  for  the  pressure  cannot  be  reduced  from 
B  to  A  when  the  engine  is  in  communication  with  the 
source,  nor  raised  from  D  to  C  when  in  communication  with 
the  refrigerator. 

33.  Conditions  of  a  reversible  cycle.  In  order 
that  a  cycle  be  reversible,  the  difference  between  the  exter- 
nal pressure  and  the  internal  during  a  change  of  volume 
must  be  infinitesimal — during  expansion  the  external  being 
infinitesimally  less,  and  during  compression  infinites.imally 
greater  than  the  internal ;  also  during  the  transfer  of  heat, 
the  difference  between  the  heat  of  the  substance  and  that  of 
the  external  body  shall  also  be  infinitesimal — during  absorp- 
tion being  infinitesimally  less  than  the  source,  and  during 
emission  infinitesimally  greater  than  the  refrigerator.  The 
differences  being  infinitesimal,  the  quantities  will  in  finite 
measures  be  equal. 

33a.  It   follows  from  the  conditions  of  the  preceding 
article,  that  if  a  closed  cycle  be  bounded 
by  the  isothermals  and  adiabatics  of  any 
substance,   the   cycle  will  be   reversible 
when  worked  with  that  substance.   Thus, 
if    there    be   an    adiabanc    compression 
along  A  -B,  Fig.    8,   an   isothermal   ex- 
pansion  along   B  A',   adiabatic    expan-  TIG  8 
sion  A'  B',  isothermal  expansion  £'  A" , 
and  so  on  back  to  A,  the  cycle  will  be  reversible. 

Also,  the  cycle  A  B  CD,  Fig.  7,  may  be  made  reversible  by 
conceiving  an  indefinite  number  of  sources  of  heat  differing  by 


THERMODYNAMICS. 


[34.] 


FIG.  9. 


d  r  and  passing  down  CD  by  an  indefinite  number  of  indefi- 
nitely short  isothermal  compressions   and  a  corresponding 
number  of  indefinitely  short  intermediate  adiabatic  expaii-  ^ 
sions  as  indicated  in  Fig.  9  ;  and  a  similar 
reversed  operation  in  ascending  from  A 

to  B. 

34.  The  heat  absorbed  by  a  sub- 
stance in  working  from  a  state  A  to  state 
B  may  be  represented  on  a  diagram  of 
energy  by  the  area  included  between  the 
path  of  the  fluid  and  the  adiabatics pass- 
ing through  A  and  B  respectively,  ex- 
tended indefinitely  in  the  direction  of  the  expansion,  Fig.  10. 
Let  A  be  the  initial  and  B  the  final  states  for  the  expan- 
sion <y,  vn  and  the  line  A  B  the  path  of  the  fluid.  Pass  the 
adiabatics  A  <p,  and  B  <pa,  then  will  the  indefinitely  extended 
area  tpl  A  B  <pt  represent  the  heat  absorbed  by  the  substance 
in  doing  the  external  work  vtAB  v»  in  the  same  units  as  vl  vt 
and  ?.\  A  ;  ihat  is,  if  v,  v,  represents 
feet,  and  v1  A,  pounds,  the  area  <p,  A 
B  cpi  will  represent  foot-pounds. 

From  the  state  B  conceive  the 
substance  to  be  expanded  adiabatically 
along  B  (p»  doing  work  as  against  a 
piston,  to  the  state  C,  then  will  the 
external  work  i\  B  C  v  have  been 
done,  without  the  absorption  or  emis- 
sion of  heat ;  and  hence  the  reduction  of  temperature  (and 
pressure)  will  be  due  to  the  transmutation  of  heat  into  work. 
At  the  constant  volume  -0  v  let  sufficient  heat  be  emitted  to 
reduce  the  pressure  to  v  D,  where  D  is  on  the  adiabatic  A  <p,; 
from  D  compress  the  substance  along  A  <p,  to  A,  during 
which  the  external  work  r,  A  D  v  will  have  been  done  upon 
the  substance ;  thence  expand  from  A  to  B  along  the  path 
A  B,  during  which  heat  must  be  absorbed.  The  only  heat 


FIG.    10. 


[34.]  THE  HEAT  ABSORBED.  23 

absorbed  in  this  cycle  of  operations  is  while  working  from 
A  to  B,  and  the  only  heat  emitted  is  in  describing  the  path 
CD;  and  since  the  cycle  is  complete  the  ideally  external 
Avork  A  B  CD  is  the  exact  equivalent  of  the  difference 
between  the  heat  absorbed  and  that  emitted,  or 
.//„  -  JIA  =  v,BCv  +  v>ABv,-vvlAD  =  AB  CD 

=  (pl  A  B  (py  —  (pl  D  C  (p» 
where  (p,  and  <p,  may  be  at  a  distance  indefinitely  great. 

Let  CD  be  moved  to  the  right  indefinitely — it  will  become 
less  and  less,  and  at  the  limit  c//"d,  or  cpt  CD  <py  will  be  zero, 
and  we  will  have 

aZTb  =  <pt  A  B  <p, 

for  the  heat  absorbed.  At  the  limit,  q>l  and  <p^  being  at  an 
indefinite  distance  to  the  right  may  be  considered  as  coincid- 
ing, and  the  path  tp^ABtp^  as  re-entrant,  forming  a  cycle,  in 
working  around  which,  heat  is  absorbed  only  along  the  path 
A  B.  The  enclosed  area  represents  what  would  be  the  exter- 
nal work  done  if  the  substance  could  be  worked  in  this 
cycle.  If  the  external  work,  v,  A  B  v,,  actually  performed 
plus  the  increased  actual  energy  of  the  substance  equals  <p, 
A  B  (p»  no  internal  work  will  be  done  in  working  along 
the  path  A  B,  but  if  these  are  unequal,  the  difference  will 
be  the  internal  work,  either  done  upon  the  substance  in 
passing  from  state  A  to  state  B,  or  by  the  substance  in  pass- 
ing from  state  Bio  A.  This  theorem  was  first  given  by  Ran- 
kine,  and  is  very  fruitful  in  the  geometrical  development  of 
this  science. 

The  heat  absorbed  in  passing  from  state  A  to  state  B  may 
be  expended  in  the  three  following  ways,  as  stated  in  arti- 
cle 9: 

1.  In  doing  the  external  work  vl  A  B  t-a  =   U ; 

2.  In  doing  internal  work  =  $; 

3.  In  increasing  the  actual  energy  of  the  substance  =  Q ; 

.-.  tp,  A  B  <?,  =  a//b  =  Q  +  S+  U.  (9) 

Any  of  the  terms  in  the  last  member  may  be  negative. 


24  THERMODYNAMICS.  [35.] 

35,  The  mechanical  equivalent  of  heat.     The 

direct  determinations  of  heat  have  been  in  terms  of  thermal 
units,  but  on  the  indicator  diagram  the  work  done  by  heat 
is  in  terms  of  foot-pounds  or  their  equivalent.  It  is  neces- 
sary to  reduce  one  of  these  to  the  other.  The  first  accurate 
determination  of  the  mechanical  equivalent  of  the  thermal 
unit  was  made  by  Dr.  Joule,  of  Manchester,  England,  who, 
after  a  series  of  experiments  beginning  in  1843  and  extend- 
ing over  a  period  of  about  seven,  years,  concluded  that  ite 
value  was  about  772  foot-pounds.  To  state  it  otherwise ;  if 
a  pound  of  water  falling  through  a  height  of  772  feet  in 
a  vacuum  should  be  suddenly  brought  to  rest,  and  all  the 
heat  thereby  generated  could  be  utilized  for  the  purpose, 
it  would  increase  its  temperature  one  degree  Fahrenheit. 
Joule's  experiments  gave  quite  a  range  of  values,  and  he  was 
inclined  to  give  more  weight  to  the  smaller  than  to  the  larger 
ones.  Later,  in  1876,  a  committee  appointed  by  the  British 
Association  for  the  Advancement  of  Science  reported  that  the 
mean  of  sixty  of  the  best  experiments  made  by  Joule  on  the 
friction  of  water  gave  774.1  foot-pounds  subject  to  a  small 
correction,  possibly  amounting  to  ^^  of  its  value,  on  account 
of  the  uncertainty  of  the  exact  position  of  the  absolute  zero 
on  the  thermometric  scale. 

Still  later,  in  1878,  Joule  made  another  set  of  experiments, 
giving  as  results  the  following  values : 

Deg.  C.  Foot-pounds, 
at  12.7,  774.6 

15.5,  773.1 

17.3,  774.0 

Mean  15.1,  773.9 

Joule's  experiments  were  made  with  water  at  about  60°  F., 
and  reducing  his  results  to  their  equivalent  for  water  at  its 
maximum  density,  according  to  the  law  indicated  by  Reg- 
nault's  experiments,  reduces  the  value  slightly,  though  for  a 


[35.J        THE   MECHANICAL   EQUIVALENT   OF   HEAT. 


25 


difference  of  20°  F.  it  will  scarcely  affect  the  first  decimal 
figure,  as  will  be  seen  when  we  consider  the  specific  heat  of 
water.  Tims,  if  the  mechanical  equivalent  of  heat  at  60°  F. 
were  774.1,  then  at  39.1°  F.,  or  say  40°  F.,  it  would  be  774 
(nearly),  and  reduced  from  the  latitude  of  Manchester  to  that 
of  New  York  it  becomes  774.8,  nearly ;  and  if  the  entire 
margin  of  error,  T^-TJ,  be  positive  and  applicable  to  this  num- 
ber, the  value  would  be  776.7,  or,  to  the  nearest  integer,  it 
would  be  777. 

More  recently,  Professor  Rowland  has  made  a  very  critical 
examination  of  the  specific  heat  of  water  at  the  lower  tem- 
peratures, and  made  a  more  accurate  determination  of  the 
mechanical  equivalent  at  those  temperatures  (On  the  Mechan- 
ical Equivalent  of  Heat,  Proc.  Am.  Acad.  of  Arts  and  Sc., 
1880).  The  most  probable  values,  as  determined  by  him, 
are,  for  the  latitude  of  Baltimore  (ibid.,  p.  196): 


Temperatures. 

Mechanical  Equivalent. 

For  Air  Thermometer. 

Mercurial 

Thermometer. 

Centigrade. 

Fahrenheit. 

Kilo.-Metres. 

Foot-pounds. 

Foot-pounds. 

o 

0 

4 

39.1 

(430.0) 

(783.4) 

(778.3) 

5 

41.0 

429.8 

783.0 

(778.1) 

6 

42.8 

429.5 

782.5 

(777.9) 

7 

44.6 

429.3 

782.2 

(777.6) 

16 

60.8 

427.2 

778.4 

(775.4) 

27 

80.6 

425.6 

775.5 

(774.3) 

36 

96.8 

425.8 

775.9 

(774.7) 

The  numbers  in  the  parentheses  we  have  computed  from 
those  of  Professor  Rowland's  tables,  the  last  column  being 
determined  by  means  of  his  table  on  page  41  of  the  Appen- 
dix to  his  paper.  It  will  here  be  seen  that  the  equivalent 
diminishes  from  39.1°  F.  to  about  80°  F.,  and,  hence,  the 


26  THERMODYNAMICS.  [35.] 

specific  heat  of  water  diminishes  according  to  the  same  law. 
Prior  to  these  experiments,  it  had  been  held,  in  accordance 
with  Kegnault's  experiments,  that  the  specific  heat  of  liquids 
increases  with  the  temperature  ;  but  according  to  the  above 
experiments  this  law  is  reversed  for  water  from  40°  F.  to 
80°  F.,  being  a  minimum  in  the  vicinity  of  the  latter  value, 
and  increasing  for  higher  temperatures.  Regnault's  experi- 
ments were  chiefly  for  higher  temperatures.  Rowland's 
values,  even  when  reduced  to  the  same  latitude,  all  exceed 
those  heretofore  used  for  scientific  and  engineering  pur- 
poses, although  they  agree  very  nearly  with  Joule's  when  re- 
duced to  the  same  thermometer,  temperature,  and  place 
(ibid.,  Appendix,  44,  45).  The  first  cause  of  difference  lies 
in  the  fact,  above  stated,  that  the  mechanical  equivalent  is 
greater  at  39°  F.  than  at  60°  F.— amounting  to  about  3 
foot-pounds — instead  of  less,  as  given  by  Regnault's  experi- 
ments. The  second  cause  is  due  to  the  fact  that  a  degree 
on  the  air  thermometer,  from  39°  to  40°,  is  perceptibly 
larger  than  on  the  mercurial  thennometer,  the  difference 
being  about  j^  of  a  degree  of  the  air  thermometer,  and  re- 
sulting in  an  increase  of  more  than  5  foot-pounds  above  that 
given  by  the  mercurial  thermometer.  Joule  used  a  mercu- 
rial thermometer. 

It  is  apparent  that  the  old  value,  772,  so  generally  used 
by  the  scientific  world,  is  much  too  small,  and  774.1,  recom- 
mended by  the  committee  of  the  British  Association,  is  not 
sufficiently  large.  According  to  Rowland's  experiments, 
the  British  Thermal  Unit  is  about  784  foot-pounds  per  de- 
gree on  the  air  thermometer,  and  nearly  779  on  the  mercu- 
rial thermometer.  Scientifically,  the  air  thermometer  should 
be  used  ;  while  for  engineering  purposes  the  mercurial  ther- 
mometer is  almost  universally  used  ;  but  in  neither  case 
should  the  highest  numbers  be  adopted  unless  the  law  of 
change  of  the  substance  be  known  throughout  the  extent  of 
the  investigation.  Such  a  law  is  not  known  with  scientific 


[36.]  FIRST  LAW   OF   THERMODYNAMICS  27 

exactness.  In  refined  analysis  it  has  been  customary  to  use 
an  empirical  formula  representing  the  experiments  of  Reg- 
nault — especially  for  water — but  in  ordinary  practice  it  is 
customary  to  consider  the  specific  heat  of  water  as  constant 
at  all  temperatures,  and  for  this  reason  it  is  not  advisable  to 
adopt  the  highest  values  given  by  experiment.  Before  de- 
ciding upon  the  value  to  be  adopted  in  this  work,  values 
were  computed  by  other  methods,  to  be  explained  hereafter, 
and  that  number  selected  which  would,  according  to  our 
present  knowledge  of  all  the  elements  involved,  harmonize 
with  the  various  methods  by  which  it  has  been  determined. 
This  number  is  778.  The  exact  value  cannot  be  found,  but, 
like  other  physical  constants,  it  may  be  determined  within 
certain  limits.  The  value  here  adopted  is  probably  within 
3-^-5-  of  its  own  value  for  the  mercurial  thermometer  at  the 
latitude  of  New  York.  The  mechanical  equivalent  we  rep- 
resent by  </,  and  call  it  Joule's  equivalent. 

36.  FIRST  LAW  OF  THERMODYNAMICS.  Heat  and  mechan- 
ical energy  are  mutually  convertible  in  the  ratio  of  about 
778  foot-pounds  for  the  British  Thermal  Unit. 

The  equivalent  in  French,  or  part  French  and  part  Eng- 
lish units,  is 

1400  foot-pounds  per  pound  of  water  per  degree 

centigrade, 

426.8  kilogramme-metres  per  kilogramme  of  water 
per  degree  centigrade. 

EXERCISES. 

1.  How  many  foot-pounds  of  heat-energy  are  there  in  one 
pound  of  coal  containing  14500  British  thermal  units  ? 

Ans. 

2.  How  far  will  one  pound  of  anthracite  coal  propel  a 
locomotive  weighing  60  tons  on  a  level  track,  friction  6 
pounds  per  ton,  if  the  entire  heat-energy  of  the  coal  could 


28  THERMODYNAMICS.  [37.] 

be   utilized   for    this  purpose,   the   coal    containing   15000 

B.  T.  U.  ? 

Ans.  miles. 

3.  If  in  melting  one  pound  of  ice  144  B.  T.  U.  become 
latent,  how  many  foot-pounds  of  energy  are  required  to 
change  the  state  of  aggregation  of  the  substance — that  is,  to 
change  ice  to  water  ? 

Ans.  112032  ft.-lbs. 

4.  Find  the  value  of  the  mechanical  equivalent  of  the 
B.  T.  U.  when  expressed  in  kilogramme-metres. 

5.  How  many  thermal  units  must  be  transformed  into 
mechanical  energy  per  minute  to  equal  one  horse-power  ? 

6.  What  is  the  theoretical  efficiency  of  a  steam  plant  that 
develops  one  horse-power  per  hour  for  every  2^  pounds  of 
coal  used,  the  heat  units  in  a  pound  of  the  coal  being  13000  ? 

7.  Steam  plants  have  been  reported  as  developing  a  horse- 
power pei  hour  with  1.5  pounds  of  coal ;  what  was  the  theo- 
retical efficiency  of  the  plant,  if  a  pound  of  the  coal  con- 
tained   15200    B.  T.  U.?     What  if    it    contained    12000 
B.  T.  U.  ? 

Ans.  In  the  latter  case,  ij  nearly. 

8.  How  many  foot-pounds  of  energy  will  be  required  to 
raise  the  temperature  of  five  pounds  of  water  from  the  tem- 
perature of  melting  ice  to  that  of  boiling  water,  the  value 
of  J  being  778  for  each  degree  ? 

9.  One  kilogramme-metre  per  degree  centigrade  equals 
how  many  foot-pounds  per  degree  Fahrenheit  ? 

Ans.     13.02. 

10.  How  many  foot-kilogrammes  of  heat  are  necessary  in 
order  to  raise  the  temperature  of  one  decigramme  of  water 
one  degree  Fahr.  ?     How  many  metre-pounds  to  raise  one 
gramme  of  water  one  degree  C.  ? 

37.  Thermal  capacities.    The  amount  of  heat  nee- 


[38.]  PRESSURE   CONSTANT.  29 

essary  to  change  by  unity  any  quality  of  unit-mass  of  a 
substance  under  given  circumstances  is  called  the  thermal 
capacity  corresponding  to  the  given  change.  Three  such  ca- 
pacities have  received  the  respective  names — specific  heat  at 
constant  pressure,  specific  heat  at  constant  volume  and  the 
latent  heat  of  expansion.  When  these  capacities  are  vari- 
able, their  values  at  a  particular  state  may  be  considered  as 
the  rate  at  which  heat  is  absorbed  per  unit  of  the  constant 
element. 

The  unit-mass,  in  English  units,  is  the  standard  one-pound 
weight,  and  in  French  imits  is  the  standard  kilogramme. 

38.  Pressure  constant.  Specific  heat  at  con- 
stant pressure.  If  the  pressure  be  constant,  the  path 
of  the  fluid  will  be  a  right  line  perpendicular  to  the  j?-axis, 
Fig.  11 ;  and  the  heat  absorbed  in  working  from  state  A  to 
state  B  along  this  line  will  be,  according  to  Article  34,  repre- 
sented by  the  area  q>^  A  B  tp,  in  foot-pounds,  to  find  which 
requires  an  experiment  with  the  substance  in  order  to  deter- 
mine its  thermal  capacity  under  constant  pressure. 

The  specific  heat  at  constant  pressure  is  the  amount  of 
heat  absorbed  in  increasing  the  temperature  of  a  unit-mass  of 
the  substance  one  degree,  the  pressure 
being  constant  and  the  specific  heat 
constant  throughout  the  degree.     In 
English   units,  it  is   the   number  of 
thermal  units  (Art.  6)  absorbed  in  rais- 
ing the  temperature  of  one  pound  of 
the  substance  one  degree  Fahrenheit. 


To   represent   it    on    a    diagram    of      °         Vl 

energy,  the  line  A  B,  perpendicular 

to  the  ^9-axis,  must  be  limited  by  two  isothermals,  as  r  and 

r  -f-  1,  differing  by  unity  of  temperature ;  then   will  the 

dynamic  specific  heat  at  and  from  r  be  represented  by  the 

indefinitely  extended  area  cpt  A  B  <p^.     If  the  specific  heat 

be  variable,  the  isothermals  must  differ  by  d  T  only. 


30  THERMODYNAMICS. 

For  numerical  values,  see  tables  of  specific  heat.  To  ex- 
press it  algebraically,  let 

/fep  =  the  ordinary  specific  heat  at  the  temperature  r  and 
pressure  p, 

Kv  =  the  equivalent  dynamic  specific  heat  ; 
then 

A;  =  jkr  (io) 

Ifdhf  =  the  thermal  units  absorbed  in  raising  the  tem- 
perature of  a  unit-mass  of  the  substance  under  a  constant 
pressure  an  amount  d  t  =  d  ?  degrees,  and  d  Hv  the  same  ex- 
pressed in  foot-pounds,  then 

dhv   =  k,dT,  (11) 

dH  =  JJevdr  =  Kvdr;  (12) 


*;  («o 

hence,  after  substituting, 


If  the  specific  heat  be  constant,  equation  (12)  integrated 
between  limits  gives 

//P  =  A,  (rt  -  r,),  (15) 

and  if  ra  —  rl  =  1,  we  have 

//p  =  A;  =  9l  A  B  <p»  Fig.  11, 
as  before  stated. 

When  the  path  is  arbitrary,  the  heat  absorbed  may  be  a 
function  of   the  three  variables^,  -y,  r,  but  when  p  or  v  is 

constant  -^  will  be  a  partial  differential   coefficient,    and 
may  be  indicated  as  above  with  a  parenthesis  and  subscript, 

7       TT 

or  with  a  parenthesis  without  a  subscript,  or  by  —  E  —  as  used 

a  r 

by  Clausius,  or  (J7t)p  as  by  M.  Saint-Robert,  or  even  with- 


r+l 


[39.J  VOLUME   CONSTANT.  31 

out  any  distinguishing  mark,  leaving  it  for  the  reader  to  infer 
its  true  character,  which  may  be  so  easily  done  in  this 
science  as  to  make  it  questionable  whether  any  mark  is 
desirable. 

39.  Volume  constant.     Specific  heat  at  con- 
stant volume  is  the  heat  absorbed  in  raising  the  tempera- 
ture of  a  unit-mass  of  a  substance  through  one  degree  when 
the  volume    is  constant,  the    specific 
heat  remaining    constant    throughout 
the  degree.     It  is  the  number  of  heat 
units  necessary  to  raise  the  tempera- 
ture of  one  pound  of  the  substance 
one  degree  F.,  the  volume  being  con- 
stant.     "When    constant   its   dynamic 
value  may  be  represented  on  a  diagram  FIG 

of  energy  by  the  area  between  a  line 
A  J?,  Fig.  12,  perpendicular   to   the 
-y-axis  limited  by  two  isothermals  differing  by  one  degree, 
and  two  indefinitely  extended  adiabatics  A  q>l  and  B  q>»  as 
shown  in  Article  34. 
Let  k,  =  the  specific  heat  for  a  constant   volume  at   the 

temperature  T  in  ordinary  thermal  units, 
KV  =  its  equivalent  dynamic  specific  heat, 

d  hv  =  the  thermal  units  absorbed  in  raising  the  tempera- 
ture d  T, 

dHv  =  the  foot-pounds  of  heat  in  d  Av  ; 
then 

d  Av  —  kv  d  r, 

=  Jkv  d  r  =  K,  d  -c  ;  (16) 


If  Kv  be  constant,  then  in  Fig.  12, 
JTV  =  <p,  A  B  <pv 


32  THERMODYNAMICS.  [40.] 


tXERCISES. 


1.  How  much  more  heat  (mechanical  energy)  is  required 
to  raise  the  temperature  of  one  pound  of  water  1°  F.  than 
of  one  pound  of  air,  the  same  amount  under  constant 


(Find  the  ratio  of  the  values  of  their  specific  heats,  as  found  from  a 
table  of  specific  heals,  and  express  their  difference  .in  foot-pounds.) 

2.  How  much  more  heat  is  required  to  raise  the  tempera- 
ture of  one  pound  of  water  1°  C.  than  of  one  pound  of  air 
the  same  amount,  at  constant  pressure  ? 

3.  How  far  must  a  mass  of  iron  fall  in  a  vacuum  in  order 
that  its  resultant  energy,   if  transmuted  into  heat,  would 
melt  it  ? 

4.  Assuming  that    air   and  hydrogen  are  perfect  gases, 
how  much  more  heat  will  be  required  to  increase  the  tem- 
perature of  one  pound  of  the  latter  one  degree  Fahr.  than 
one  pound  of  the  former  the  same  amount  (     Express  the 
difference  in  thermal  units  and  in  foot-pounds. 

5.  If  oxygen  and  hydrogen  are  perfect  gases,  how  many 
pounds  of  oxygen  will  be  required  in  order  to  contain  as 
much  heat  as  one  pound  of  hydrogen  at  the  same  tempera- 
ture ? 

6.  Show  that  a  right  line  parallel  to  the  v-axis  will  be 
divided  into  equal  parts  by  a  series  of  isothermals  of  which 
the  general  equation  is^>  v  =  R  r,  provided  ra  —  TI  =  T3  —  ra, 
&c. 

7.  In   the   preceding  exercise,   show  that   a   right   line 
parallel  to  the  j>axis  will  also  be  divided  into  equal  parts. 

8.  Show  that  a  line  drawn  through  the  origin  of  co-ordi- 
nates is  not  divided  into  equal  parts  by  the  successive  equi- 
lateral hyperbolas  of  Exercise  6. 

4O.  Let  the  temperature  be  constant  during 
expansion.     SECOND  LAW.     In  this  case  the  path  of  the 


[40.  J  TEMPERATURE  DURING  EXPANSION.  33 

fluid  will  be  an  isothermal,  as  A  B,  Fig.  13  ;  and  the  heat 
absorbed  during  the  expansion  from  vl  to  -y,  will,  according 
to  Article  34,  be  represented  by  the  area  <pl  A  B  cpn  bounded 
by  the   isothermal  A  B   and   the   two 
adiabatics~^l  (pl  and   B  tp,  indefinitely 
extended.     If  H^  be  the  heat  absorbed,          %j£$Ss£^  B 
the  subscript  indicating  that  the  entire 
heat  absorbed  is  to  be  at  one  tempera- 
ture ;  then 

fft  =  cp.AB  cp.. 

o     -v, 

Since  the  temperature  of  the  work-  FIG.  13. 

ing  substance,  in  this  case,  is  uniform 
during  expansion,  it  is  assumed  that  the  actual  energy  of 
the  working  substance  remains  unchanged,  and  hence  Q  in 
equation  (9)  will  be  zero,  and  we  have 


That  is,  during  isothermal  expansion  the  heat  absorbed 
equals  the  entire  work  done,  both  external  and  internal.  This 
heat  cannot  be  directly  measured,  but  it  may  be  computed, 
as  will  appear  from  this  and  the  two  following  articles.  No 
engine  can  transmute  into  external  work  all  the  heat  absorbed 
by  the  working  substance,  some  of  it  being  always  rejected 
at  a  lower  temperature  than  the  source.  Experience  confirms 
the  following  principle,  called  the 

SECOND  LAW.  If  all  the  heat  absorbed  be  at  one  temper- 
ature, and  that  rejected  be  at  one  lower  temperature,  then 
vnll  the  heat  which  is  transmuted  into  work  be  to  the  entire 
heat  absorbed  in  the  same  ratio  as  the  difference  between  the 
absolute  temperatures  of  the  source  and  refrigerator  is  to 
the  absolute  temperature  of  the  source.  (Appendix  ii.) 

In  other  words,  the  second  law  is  an  expression  for  the 
efficiency  of  the  perfect  elementary  engine. 

The  object  of  the  second  law  is  to  furnish  a  basis  for  the 


34  THERMODYNAMICS.  [40.] 

computation  of  the  heat  absorbed  during  expansion,  the 
heat  of  the  working  fluid  being  maintained  at  a  constant 
temperature.  Thus,  if  between  600°  F.  and  500°  F.  ab- 
solute, in  a  perfect  elementary  engine,  ten  thermal  units  be 
transmuted  into  work,  then  will  the  heat  absorbed  at  600°  F. 
have  been 

h  =  10 IL_    =  60 


thermal  units.  If  the  expansion  be  isothermal,  the  equation 
to  the  path  of  the  fluid  will  be  that  of  the  gas  at  constant 
temperature,  and  the  external  work  may  be  directly  com- 
puted, from  the  equation  to  the  gas,  being  represented  in 
Fig.  13,  by 

IT  =  v,  A  B  v,  =  fpdv. 

It  will  be  observed  in  Fig.  13  that,  if  the  area  (p^ABcp^ 
be  divided  into  an  indefinite  number  of  strips,  representing 
Carnot's  cycles,  ultimately  the  topmost  strip  A  B  c  d  will 
equal  the  topmost  strip  of  v,  A  B  v»  representing  external 
work  cut  off  by  the  second  isothermal.  If  the  work  done 
in  those  Carnot's  cycles  be  equal,  the  total  heat,  <plAB  <?„ 
will,  according  to  the  second  law,  be  the  area  of  the  top- 
most one  multiplied  by  the  number  of  cycles.  The  top- 
most one,  Fig.  13,  will  be  the  differential  of  the  external 
work,  or  d  TF,  and  rt  —  rt  becomes  d  T,  and  if  TI  be  the 
constant  temperature  at  which  heat  is  absorbed,  which  will 
'je  the  temperature  of  the  isothermal  A  J3,  we  have 


The  second  law  is  the  result  of  observations,  experiments, 


[40.]  SECOND    LAW.  35 

and  deduction.  It  is  not,  like  a  proposition  in  geometry, 
capable  of  a  direct,  rigid  demonstration ;  but  rather,  like  the 
axioms  of  geometry,  appeals  to  our  understanding  for  as- 
sent when  the  terms  used  and  the  operations  assumed  are 
well  understood.  Or,  perhaps  a  better  parallel  will  be 
found  in  the  Newtonian  laws  of  motion,  which  were  first 
conceived,  from  the  results  of  experiments,  to  represent 
ideally  perfect  conditions,  and  later  became  firmly  estab- 
lished by  the  fact  that  when  applied  to  the  solution  of  prob- 
lems in  nature  the  results  obtained  agreed  with  those  ob- 
served. So  this  law,  first  conceived  to  represent  what  would 
be  the  results  of  experiments  if  the  conditions  were  perfect, 
has  become  firmly  established  through  the  fact  that  it  has 
successfully  stood  the  many  crucial  tests  to  which  it  has 
been  subjected.  If  the  formulas  founded  upon  it  had  led 
to  results  known  to  be  erroneous,  they  would  have  dis- 
proved the  law ;  but  it  has  been  found  that  all  the  results  so 
deduced  agree  with  those  of  experiment  at  least  within 
the  limits  of  the  errors  of  observation. 

Carnot  made  the  first  step  toward  the  establishment  of 
the  law  by  showing  that  the  efficiency  of  his  ideal  engine 
was  a  direct  function  of  the  difference  of  the  temperatures 
of  the  source  and  refrigerator,  and  was  independent  of  the 
nature  of  the  working  fluid.  The  idea  of  an  absolute  tem- 
perature had  not  then  entered  this  science.  Later  the  law 
became  established  through  the  labors  of  Clausius,  and  of 
Joule  and  Thomson.  Rankine  virtually  deduced  it  from 
his  theory  of  molecular  vortices.  He  came  to  the  conclusion 
that  Oarnot's  law  is  not  an  Independent  principle,  but  is  de- 
ducible  from  the  equations  of  the  mutual  conversion  of  heat 
and  expansive  power. 

Let  one  pound  of  any  substance  having  the  volume  0  va 
pressure  vl  A,  and  absolute  temperature  T,  in  constant  com- 
munication with  a  source  of  heat  at  the  same  temperature 
(or  at  a  temperature  r  -j-  d  T),  expand  from  -y,  to  vy  by  driv- 


36  THERMODYNAMICS.  [40.J 

ing  a  piston  in  a  cylinder ;  then  will  the  indefinitely  ex- 
tended area  (p,  A  B  (p.,,  expressed  in  foot-pounds,  represent 
the  heat  absorbed,  and  divided  by  778  will  be  the  value  of 
the  heat  in  British  thermal  units. 

To  find  the  area  ^AB  q>n  conceive  it  to  be  divided  into 
an  indefinite  number  of  strips  of  equal  areas  by  isothermals 
of  the  given  substance,  as  dc,jit<...y  z,  &c. ;  they  will  repre- 
sent equal  quantities  of  heat,  and  if  an  elementary  engine 
be  worked  in  the  successive  cycles  A  B  c  d,  d  c  ij,  &c.,  the  re- 
sultant works  done  will  also  be  equal.  These  are  Carnofs 
cycles,  since  all  the  heat  absorbed  will  be  at  one  temperature, 
and  that  which  is  rejected,  at  one  lower  temperature.  Let 
the  successive  equal  quantities  of  heat  thus  transmuted  into 
external  work  constitute  a  scale  of  temperatures — known 
as  Thomson's  Absolute  Thermometric  Scale  (Phil.  May.,  xi. 
(1856)  '216.  Thomson's  Papers,  p.  100).  The  characteristic 
quality  of  this  scale  is  —  equal  quantities  of  heat  when 
worked  in  Carnot's  cycle  will  do  equal  quantities  of  external 
work  'Independently  of  the  nature  of  the  working  substance. 
At  first,  any  amount  of  heat  or  area,  as  ij  d  c,  may  be  taken, 
arbitrarily,  as  a  unit,  and  a  repetition  of  this  unit  will  con- 
stitute a  scale  of  natural  numbers,  as  7,  8,  9,  tfcc.,  the  zero 
of  which  may  be  placed  arbitrarily.  Having  assigned  its 
place  and  the  unit  of  heat,  the  quantity  of  heat  involved  in 
any  number  of  such  operations  becomes  known.  Thus,  the 
heat  necessarily  destroyed  in  performing  the  operations 
numbered  7,  8  and  9  will  be  three  times  the  unit  initially 
assumed.  Fractional  parts  of  the  scale  will  correspond  to 
fractional  parts  of  the  unit.  The  scale  may  be  so  numbered 
that  the  two  fixed  points  shall  correspond  with  32°  F.  and 
212°  F. 

Conceive  that  the  zero  of  the  scale  corresponds  with  the 
total  deprivation  of  heat  from  the  substance,  and  that  in 
raising  the  pressure  from  0  to  t\  A,  there  are  T  of  the 
arbitrary  units.  Let  each  unit  be  divided  into  an  indefinite 


[40.J  TEMPERATURE   DURING   EXPANSION.  37 

number  of  equal  parts  by  isothermals,  each  represented  by 
d  r  ;  then  will  the  number  of  parts  in  each  unit  be  1  -f-  d  T, 
and  the  number  in  r  units  will  be 

number  of  strips  =  —  • 

The  area  of  any  one  of  the  infinitesimal  strips,  as  A  B  c  d, 
being  known,  we  have 

tp,  AB<p,  =  —  x  A  Bed,  (18) 

dr 

and  the  solution  is  now  reduced  to  that  of  finding  the  area 
A  B  c  d.  Conceive  it  to  be  divided  into  an  indefinite 
number  of  parts  by  vertical  lines  having  between  them  the 
constant  abscissa  d  v  (or,  more  accurately,  let  the  divisions 
be  made  by  adiabatics  having  between  their  upper  ends  the 
abscissa  d  v\  then 

efhg  =  dpdv, 
and 

A  B  c  d  =  2^  dp  d  v. 

This  summation  cannot,  generally,  be  performed  by  an 
integration,  for,  generally,  dp  varies  from  A  to  B,  and  is 
not  simply  a  function  of  v.  For  any  assigned  value  of  r, 
dp  depends  directly  upon  d  r,  since  it  is  limited  by  two  con- 
secutive isothermals  differing  by  d  r,  a  condition  which,  in 
the  language  of  the  calculus,  is  indicated  by  the  expression 

ae=CM\dr, 


thus,  changing  from  dp  independent  to  p  dependent  upon 
r.      Substituting  this  for  dp  above,  gives 


A.  £  o  d  = 

j  ^\ 

where  d  r  is  placed  outside  the  integral  sign  since  it  is  con- 


38  THERMODYNAMICS.  [40.] 

stant  throughout  the  strip  A  B.    This  value  in  equation  (18) 


^nABv^r       V*          d  v. 

t/V  ! 

If  the  expansion  be  infinitesimal  and  equals  dv,  we  have 


which  is  the  required  equation. 

In  using  this  equation,  (-~\   is  to  be  found  by  differen- 
\d  tj 

tiating  the  equation  of  the  gas,  given  in  terms  of  the  abso- 
lute scale  of  temperatures,  considering  v  as  constant  ;  or  by  an 
experiment,  finding  the  change  of  pressure  for  a  very  small 
change  of  temperature,  but  in  integrating  from  vl  to  vt  the 
temperature  must  be  constant,  so  that  not  only  will  T,  if 

any  in  (  ~r"\  be  constant,  but  the  T  before  the  integral  sign 

will  also  be  constant.  Indicate  this  by  r,.  Since  an  amount  of 
heat  equal  to  that  absorbed  by  any  substance  during  isother- 
mal expansion  becomes  latent,  the  preceding  equation,  more 
completely  expressed,  becomes  : 

LATENT    HEAT    OF  |  /     '17/7    \ 

EXPANSION  from  J-  =  //T=  r,    /        ('l/')    dv  (21) 

«,*>«,  JVi      £**'*     J^ 

in  which  the  subscript  r,  of  the  bracket  implies,  as  explained 
above,  that  r  within  the  parenthesis  is  to  be  considered  con- 
stant during  the  integration.  As  a  thermal  capacity,  Article 

37,  the  latent  heat  of  expansion  is  r  (—P  as  if  d  v  were  unity, 

dr 

being  the  rate  at  which  heat  is  absorbed  per  unit  of  volume. 
Differentiating  equation  (21),  considering  r  as  constant  and 
v  variable,  gives 


[40.]  TEMPERATURE   DURING   EXPANSION.  39 

which  is  the  same  as  equation  (20),  where  d  v  is  the  abscissa 
of  b  in  reference  to  v^,  Fig.  13. 

'The  scale  of  temperatures  above  used  is  not  practical,  except 
for  the  purposes  of  analysis,  since  heat  cannot  be  actually 
divided  with  accuracy  by  any  known  means  according  to  the 
process  described  ;  and  it  remains  to  be  shown  how  the  re- 
sult can  be  made  of  practical  value.  Conceive  a  quantity  of 
heat  equal  to  that  absorbed  by  a  pound  of  the  substance, 
cp^AB  (p»  to  be  absorbed  by  such  a  quantity  of  &  perfect  gas 
as  will  give  the  same  temperature,  and  let  the  temperatures 
be  measured  by  an  ideally  perfect  gas  thermometer  graduat- 
ed from  absolute  zero  and  having  T  equal  divisions  up  to  the 
temperature  here  considered  ;  then  will  equal  divisions  on 
this  scale  correspond  with  equal  quantities  of  actual  heat  in 
the  perfect  gas — so  that,  if  the  gas  be  cooled  by  abstracting 
equal,  successive  quantities  of  heat,  the  successive  tempera- 
tures will  be  indicated  by  equal  divisions  on  the  scale.  In 
this  manner,  the  heat  in  a  perfect  gas  might  be  divided  into 
equal  parts.  Let  the  temperature  of  the  given  substance  be 
reduced  an  amount  d  r  on  this  scale  by  working  the  heat  in 
Carnof  s  cycle,  the  same  amount  of  heat  will  be  transmuted 
into  work  as  must  be  abstracted  from  the  perfect  gas  in  re- 
ducing its  temperature  the  same  amount,  and  so  on.  Con- 
ceive isothermals  of  the  substance  to  be  drawn  on  the  dia- 
gram of  energy,  differing  by  d  t  of  the  perfect  gas  thermom- 
eter ;  there  will  be  *  -f-  d  t  such  divisions  between  zero  and 
r,  as  in  the  former  case.  These  isothermals  may  be  con- 
ceived to  be  described,  geometrically,  from  the  equation  of 
the  gas  given  in  terms  of  the  scale  of  the  perfect  gas  ther- 
mometer, or,  physically,  by  supplying  heat  to  the  expanding 
gas  so  that  the  temperature  will  remain  constant  as  indicated 
by  this  thermometer  and  noting  the  contemporaneous  press- 
ures and  volumes.  These  processes,  perfectly  done,  would 
give  the  same  isothermals ;  and  since  the  number  is  made 
the  same  as  in  the  earlier  part  of  this  article  where  the  strips 


40  THERMODYNAMICS.  [40.] 

were,  arbitrarily,  'made  equal,  and  since  the  lowest,  or  zero- 
isothermals,  coincide,  also  the  highest,  or  r-isothermals,  it  is 
inferred  that  the  successive  isothermals  in  the  two  cases  co- 
incide. It  follows,  then,  that  if  the  area  ^ABtp^  be  inter- 
sected by  isothermals  differing  by  an  absolute  constant  tem- 
perature, the  areas  between  the  successive  isothermals  will  be 
equal;  and  if  the  number  representing  the  difference  of 
temperatures  be  commensurable  with  the  number  represent- 
ing the  highest  temperature,  the  entire  area  (p^ABcp^  will  be 
divided  into  equal  parts.  By  making  the  difference  indefinite- 
ly small,  or  d  T,  the  question  of  commensurability  disappears. 

But  a  perfect  gas  is  unknown  ;  it  has,  however,  been  found, 
as  stated  in  Articles  14  and  16,  that  the  air  thermometer  dif- 
fers but  little  from  that  of  a  perfect  gas  thermometer, 
the  temperature  of  melting  ice  being  491.13°  F.  above  the 
absolute  zero  of  the  air  thermometer,  and  about  492.66°  F. 
above  the  zero  of  the  absolute  scale,  a  difference  of  about 
•sfa  of  the  entire  491°,  a  quantity  too  small  to  be  measured 
in  actual  practice,  and  can  be  determined  only  by  the  most 
refined  experiments.  The  position  of  the  zero  of  the  abso- 
lute scale  cannot  be  determined  exactly,  but,  accepting  the 
results  of  Thomson  and  Joule,  if  the  zero  of  the  air  ther- 
mometer be  made  to  coincide  with  the  melting  point  of  ice, 
then  by  adding  492.66°  F.  to  the  reading  of  the  air  ther- 
mometer, the  sum  will  be  the  value  of  the  temperature  on 
the  absolute  scale,  almost  exactly. 

Equations  (20)  and  (21)  are  theoretically  exact,  and  hence 
are  practically  so  for  volumes,  pressures  and  temperatures 
determined  by  the  best  methods  known. 

The  following  reasoning  may  aid  the  reader  in  satisfying 
himself  of  the  equality  of  the  strips.  Conceive  the  area 
<p,AB  <?„  Fig.  14,  to  be  divided  into  an  indefinite  number  of 
strips  by  isothermals  of  the  substance,  differing  by  the  con- 
stant absolute  temperature  d  T,  then  will  the  areas  thus 
formed  be  equal. 

If  the  areas  between  equidistant*sothermals  and  the  adia 


[41.] 


TEMPERATURE   DURING   EXPANSION-. 


41 


FIG.   14. 


batics  are  not  equal,  a  line,  £  N,  may 
be  so  drawn  that  they  will  be  equal, 
but  the  area  <pt  A  B  N  (Fig.  14)  will 
be  an  area  exceeding  that  which  rep- 
resents the  heat 
absorbed ;  or,  if 
it  falls  within 
B  q>^  it  will  be 
less  than  that 
representing 
the  heat  absorbed. 

If  the  working  fluid  is  a  perfect 
gas  the  areas  abed  and  efg  h  (Fig. 
15)  will  be  equal,  but  if  the  gas  be  imperfect  all  the  small 
areas  in  <p,  A  £  tp,  below  the  topmost  one  will  exceed  those 
between  the  corresponding  isothermals  in  vl  A  B  vt. 

41.  Thomson  and  Joule  established  equation  (20)  upon  a 
principle  of  Carnot.  Carnot  proved  that,  of  the  heat  ab- 
sorbed, 9>,  A  b  n,  Fig.  13,  during  isothermal  expansion,  the 
part  d  b  transmuted  into  work  by  working  in  one  of  his 
cycles,  was  ^  (rl — T^)  of  the  heat  absorbed,  where  /*  is  a  func- 
tion of  the  higher  temperature  only  and  hence  independent 
of  the  nature  of  the  working  substance,  and  r,  —  Ta,  the 
fall  in  temperature  of  the  working  substance.  In  this  case, 
let  rl  —  r2  =  d  r,  then  will  //  d  r  be  the  fractional  part  of 
fpl  A  b  n  transmuted  into  work. 

Let  J^be  the  latent  heat  of  expansion  in  thermal  units, 
then  will  M  d  v  be  the  heat  units  in  q>tA  b  n,  and  in  foot- 
pounds we  have 

J  M  d  v  =  (pl  A  b  n, 

and  the  heat  transmuted  into  the  work  b  d  will  be 

b  d  =  jn  J  M  d  v  d  T. 
But  we  also  have 

bd= 


42  THEEMODYNAMICS.  [41.] 

and  making  these  equal  gives 

SJf=l(*J>\ 

j*  \drJ 

Carnot  did  not  find  the  form  of  the  function  //,  In  re- 
gard to  it  Thomson  says :  "  It  has  an  absolute  value,  the  same 
for  all  substances  for  any  given  temperature,  but  which  may 
vary  with  the  temperature  in  a  manner  that  can  only  be 
determined  by  experiment"  (Thomson's  Papers,  p.  187). 
Thomson,  whose  resources  ever  seem  sufficient  for  the  oc- 
casion, set  about  its  determination,  the  processes  for  which  are 
described  in  the  Philosophical  Magazine,  and  more  recent- 
ly in  Thomson's  Mathematical  and  Physical  Papers,  cover- 
ing many  pages.  Early  in  the  investigation,  Joule  suggest- 
ed that  the  value  of  n  might  be  "  inversely  as  the  tempera- 
ture from  zero"  (Thomson's  Papers,  p.1  199);  and  these 
experimenters  established  the  truth  of  this  suggestion  by 
that  celebrated  series  of  experiments  known  as  "  the  experi- 
ments with  porous  plugs."  Hence,  we  have 

p  =  -; 

r 


as  already  found.  The  quantity  /*  is  known  as  "  Garnet's 
function,"  the  title  given  to  it  by  Sir  William  Thomson.  The 
value  1  -s-  /i  =  TOT  the  absolute  temperature  of  melting  ice, 
was  found  to  be  273.68°  C.  (ibid.,  p.  391). 

Thomson's  absolute  scale  may  be  thus  defined :  The  num- 
bers expressing  degrees  of  absolute  temperatures  are  propor- 
tional to  the  quantities  of  heat  absorbed  and  emitted  at  those 
temperatures  in  a  reversible  cycle.  Thus,  if  //  =  g>t  A  B 
<#,  =  the  heat  absorbed,  Fig.  13,  and  it  be  divided  into  r 
equal  parts,  then  will  one  part  be  H  -j-  r;  and  if  h  be  the 


[42.]  TEMPEEATUEE   DUEING   EXPANSION.  43 

heat  emitted  =  gjt  y  z  cp^  then  ;will  the  number  of  equal 
parts  in  h  be 

h  +  (H+  T)  =  ~T  =  t(^y); 

h   _   t 
'"H  ~  7' 

The  equal  parts  of  heat  in  qj^ABcp^  may  be  conceived 
to  be  secured,  physically,  by  a  succession  of  perfect  engines 
in  which  the  refrigerator  of  one  is  the  source  of  the  next, 
and  so  on.  It  was  in  this  manner  that  Carnot  established 
his  expression  for  efficiency.  The  amount  of  work  done  by 
heat  depended  only  upon  the  difference  of  the  temperatures 
of  the  source  and  refrigerator  and  some  function  of  the 
higher  temperature,  as  already  given. 

4:2.  To  express  equation  (21)  in  terms  of  the  external 
work,  from  Fig.  13,  we  have 

U  =  v,  A  B  vz  =    F  p  d  v  ; 

.  •.  dU  =•  p  d  v  ; 


.-.*?  =  (*£)  d  v,  also  written    *  d  U', 
dr    '     \d  rJ  dr 

hence,  substituting  in  (21), 

9lAB  <p»  Fig.  13  ;  (22) 


, 

dv  /  d  T 

From  equation  (22)  it  appears  that  the  heat  absorbed  may 
be  found  from  the  temperature  at  which  it  is  absorbed  and 


44  THERMODYNAMICS.  [43.] 

the  external  work  regarded  as  a  function  of  the  tempera- 
ture. 

The  three  preceding  articles  will  be  more  clearly  under- 
stood after  becoming  more  thoroughly  familiar  with  the  sub- 
ject as  developed  in  the  following  pages. 

43.  In  equation  (19)  the  quantity  under  the  integral  sign 
for  a  given  expansion  is  constant,  hence  d  T  may  be  inte- 
grated between  limits,  giving 


(23) 

of  Fig.  13,  in  which  rt  is  the  absolute  temperature  of  the 
isothermal  A  B  and  ra  that  of  y  z,  and  //,  the  heat  absorbed 
along  A  B  and  7/a  that  rejected  along  z  y.  Heat  absorbed 
during  an  operation  may  be  considered  positive  and  that 
emitted,  negative. 

EXERCISES. 

1.  If  the  equation  of  the  gas  bep  v  —  R  T,  find  the  heat 
absorbed   during  expansion  at  the  constant  temperature  of 
500°  F.    from   vt  =  ten  cubic  feet  to  v,  =  30  cubic  feet. 
(Use  Eq.  (21).) 

2.  If  the  equation  of  the  fluid  be  »  =  R  -  -  -  _  (of 

V  TV* 

which  carbonic  acid  gas  is  a  special  case),  find  the  area  of 
one  of  the  strips  in  Fig.  13  for  a  difference  of  temperature 
d  T,  for  an  expansion  from  vt  to  ?'„  at  the  temperature  T,. 

Ans. 


.  [*  &*.«•  +  J.  (L  -LV]  Jr. 

•«,       T*  \v,       r,/J 

3.  Find  the  latent  heat  of  expansion  in  the  preceding  ex- 
ercise. 

4.  If  the  equation  of  the  gas  be  p  —  R  -  —  _  _,  find 

v         TV* 


[44.]  GENERAL   CASE.  45 

the  external  work  done  in  expanding  from  v1  to  -y,  at  the 
temperature  r..  , 

Ans.   Brjoff^-1'-  (1-1). 

6  V,  Tl   \V1          vj 

5.  Find  the  internal  work  done  in  exercise  4. 

6.  In  exercise  4  find  the  area  between  the  two  consecutive 
adiabatics  A  (p,  and  b  n,  Fig.  13. 

7.  In  exercise  4  find  the  ratio  of  the  internal  work  to  the 
external  for  an  expansion  from   vt  —  9  to  vy  =  18   cubic 
feet,  at  T  =  700°  F.,  £  =  35,  and  b  =  48000. 

8.  If  the  equation  of  a  gas  werep  =  4  v  r,  find  the  heat 
absorbed   at  the   temperature  T  =.  600°    F.   in   expanding 
from  20  to  30  cubic  feet,  and  reduce  to  thermal  units. 

44.  General  Case. — Let  the  path  of  the  fluid  be 
arbitrary,  as  A  -B,  Fig.  16,  A  <?„  B  <p2,  two  adiabatics  in- 
definitely extended  to  the  right,  then,  as  shown  in  Article  34, 
the  area  <p,  A  B  cp^  will  represent  the 
heat  absorbed  in  passing  from  state 
A  to  state  B.  To  find  this  area,  con- 
ceive it  to  be  divided  into  an  indefi- 
nite number  of  indefinitely  narrow 
strips,  as  follows: — Divide  the  line 
A  B  into  an  indefinite  number  of  parts 
by  the  isothermals  a  o,  bp,  &c.,  differ- 
ing by  d  T,  the  points  of  division  being 
at  a,  b,  c,  &c. ;  and  from  these  points  draw  verticals  inter- 
secting the  isothermals  next  below  in  the  points  n,  o,p,  q,  &c. 
Through  the  points  «,  b,  c,  &c.,  and  ??,  o,  jt?,  &c.,  draw  adi- 
abatics, as  a  m,  o  m^  b  ms ;  then  will  the  sum  of  all  the  strips 
m  a  o  m»  m,  o  b  m3,  &c.,  ultimately  equal  the  area  q>,  A  B  q>y. 
If  r  be  the  absolute  temperature  of  any  isothermal,  as  a  o, 
and  d  v  the  expansion  from  state  a  to  state  o,  then,  accord- 
ing to  equation  (20),  will  the  area 

m  a  o  mt  =  r  [vM  d  v ; 


46  THERMODYNAMICS.  [45.] 

and,  according  to  Article  39,  the  area 

my  o  b  m^  =  KV  d  r  ; 
hence,  ultimately, 

d  «,   (24) 


which  is  a  GENERAL  differential  EQUATION  OF  THERMODY- 
NAMICS. 

In  this  solution  the  polygon  A  n  a  o  bp,  &c.,  is  inscribed 
in  the  figure  <pl  A  B  <p»  but  the  same  result  would  be 
reached  if  the  polygon  were  circumscribed,  as  indicated  in 
the  figure  by  A  u  a  w  J,  &c. 

From  equation  (24)  we  have 


9lAB^  =  A//B  =  'K,  d  r  +  f  dv,   (25) 

but  the  general  integral  cannot  be  found  since  A"v  is  not  a 


known  function  of  T.  nor  T  and  \-         known  functions  of 

\d  tl 

v.  In  equation  (24)  r  and  v  are  independent  variables. 
The  shaded  strip  m  a  o  mt  represents  heat  transmuted  into 
work  due  to  an  isothermal  expansion,  and  the  unshaded 
strip  my  o  b  m%  the  increased  energy  of  the  substance,  both 
actual  and  potential,  due  to  the  change  of  temperature  in 
passing  from  a  to  b. 

45.  To  make  r  and  p  inde- 
pendent variables.  Intersect  the 
path  A  £,  Fig.  17,  with  consecutive 
isothermals  differing  by  the  constant 
d  T,  as  before  ;  and  from  the  points  of 
division  5,  c,  d,  &c.,  draw  lines  parallel 
to  the  axis  O  v,  intersecting  the  ad- 
17.  jacent  isothermals  in  the  points  n,  o, 

<fec.,  thus  describing  an  inscribed  poly- 
gon, A  n  b  o,  &c.  A  circumscribed  polygon  would  answer  the 
same  purpose.  Through  the  vertices  of  the  polygon,  A,  nt 


[45.]  r  AND    P   INDEPENDENT   VARIABLES.  47 

&,  o,  c,  &c.,  draw  the  adiabatics  b  mlt  c  ma,  o  m3,  &c.,  then, 
ultimately, 

m1b  c  mt  =  d  If  =  m^  o  m3  —  ?nt  c  o  ma 

=  Kv  d  r  —  raa  c  o  mt)  (Eq.  (12)), 


|J   di>,(Eq.  (20)). 

But  <7  v  in  this  equation  is  the  abscissa  of  o  in  reference 
to  c  (Figs.  17  and  18\  on  an  isother- 
mal through   c,  and  hence  is  not  de- 
termined  directly  from  the  equation 
of  the  path.     Change  this  d  v  to  d  vf       & 
and  let  b  k  =  d  v,  being  the  differ- 
ence of  the  abscissas  of  two  consecu- 
tive points  of  the  path ;  then 

—  '    —  the  rate  of  change  of  pressure, 


f 

(dp\  ,  , 

(  -j£-  ]  dr  =  ()l  =  cv  =  increase  01  pressure, 

-J  drdv'  =  are&icoj,         (-^ — J    dT  =  bo, 


But  icoj  =  l)oce  having  the  common  base  c  o  and 
between  the  same  parallels.  Multiply  ing  the  last  expression 
by  r  -f-  d  T,  and  substituting,  gives 

dH=Kvdr-r(d-^]dp.  (26) 


Equation  (26)  is  a  second  GENERAL  EQUATION  of  thermody- 
namics in  which  r  and  p  are  the  independent  variables. 
Other  forms  may  be  deduced  from  these,  as  will  hereafter 
be  shown.  For  convenience  of  reference  these  equations 
are  brought  together. 


48  THERMODYNAMICS.  [46.] 

46.  The  two  fundamental  equations  of  ther- 
modynamics are  : — 

d  JI  =  Kv  d  T  -|-  T  I  ~£\  d  v. 

(A) 


The  remainder  of  this  work  will  consist  chiefly  of  a  dis- 
cussion of  these  equations.  Thermodynamics  is  the  science 
which  treats  of  the  mechanical  theory  of  heat. 

QUESTIONS  FOR  EXAMINATION. 
(Some  of  these  questions  require  knowledge  outside  of  this  text.) 

Give  instances  of  heat  generated  by  mechanical  action.  Draw  infer- 
ence. What  did  Count  Rumford  conclude?  Describe  Davy's  experi- 
ment. Is  his  experiment  conclusive  ?  Who  first  made  an  exact  deter- 
mination of  the  mechanical  equivalent  ?  Describe  his  methods.  How 
long  did  he  devote  himself  to  the  subject  ?  What  did  he  consider  the 
most  probable  value  ?  What  is  meant  by  work  ?  momentum  ?  energy  ? 
foot-pound  ?  horse-power  ?  metre-kilogram  ?  heat  unit  ?  caloric  ?  rate? 
Define  exactly  the  "  thermal  equivalent."  Why  does  the  mechanical 
equivalent  depend  upon  latitude  ?  altitude  ?  thermometer  used  ?  In 
what  respects  do  the  results  of  Rowland's  experiments  differ  from  Reg- 
nault's  ?  Is  perpetual  motion  possible  ?  Why  not  ?  When  a  gas  ex- 
pands, why  does  the  temperature  fall  ?  When  it  expands  into  a  vacuum, 
does  its  temperature  fall  ? 

What  is  an  atom  ?  molecule  ?  "  ether  "?  What  are  he  it  rays  ?  In  what 
respects  do  heat  and  light  differ  ?  agree  ?  When  is  a  body  transparent  ? 
op:ique  ?  athermauous  ?  diathermanous  ?  When  is  a  body  heated  by 
radiation  ?  conduction  ?  When  a  body  is  heated,  what  three  effects  may 
be  produced  ?  What  is  specific  volume  ?  specific  pressure  ?  specific  grav- 
ity ?  specific  heat  ?  real  specific  heat  ?  apparent  specific  he  >.t  ?  latent 
heat  ?  latent  heat  of  expansion  ?  thermal  capacity  ? 

What  is  a  perfect  gas  ?  imperfect  gas  ?  Does  the  coefficient  of  expao- 
sion  vary  with  different  gases  ?  For  what  is  it  least  ?  What  is  the  "  ab- 
solute zero"?  Can  it  be  realized  ?  Of  what  value  is  it  in  theory'; 
What  is  thermodynamics  ?  What  is  a  general  equation  of  thermodynam- 
ics ?  Eliminate  dr  from  equations  (A),  and  deduce  a  third  equation  for 
dH. 


CHAPTER  II. 

PERFECT       GASES. 

47.  Difference   of    specific    heats.      From  the 
equation  of  a  perfect  gas,  equation  (2),  we  find 

(*£}=*•=•£., 

\d  T  )  V  T  ' 

(dv  \  =   R^  __  v_ 
\dr)         p    "    T  ' 

and  these  in  equations  (A)  give 

dll=  Kvdr  -\-pdv. 


From  equation  (2)  j?  d  v  +  v  d  p  =  Rdr,  which  in 
equation  (27),  after  placing  the  second  members  equal, 
give 

K^  —  K,  =  E  ;  (28) 

hence,  the  difference  of  the  two  specific  heats  for  a  perfect 
gas  is  constant. 

48.  Specific  heat  constant.  In  a  perfect  gas  no 
internal  work  is  done  during  a  change  of  state,  hence,  at 
constant  volume,  no  work  will  be  done  by  the  absorption  of 
heat,  and  all  the  heat  absorbed  will  be  sensible  at  all  temper- 
atures ;  hence,  the  specific  heat  of  a  perfect  gas  at  constant 
volume  will  be  constant,  and  equation  (28)  shows  that,  in 
this  case,  the  specific  heat  at  constant  pressure  will  also  be 
constant.  It  is  found  that  the  specific  heat  for  sensibly  per- 
fect gases  at  constant  volume  is  independent  of  the  volume. 


60  PERFECT   GASES.  [48.] 

Let  JTV  =  Cv  and  Kv  =  <7P  for  sensibly  perfect  gases,  and 
equations  (27)  become 


which  are  the  general  equations  of  sensibly  perfect  gases. 
Equation  (28)  becomes 

R  =  C\  -  Cv.  (29) 

When  Clausius  first  established  the  preceding  equation, 
he  concluded  that  both  specific  heats  were  constant  for  per- 
fect gases  at  all  pressures  and  temperatures,  although  this 
view  opposed  the  one  then  prevalent  —  that  the  specific  heat 
was  a  function  of  the  density  of  the  gas.  Soon  after,  how- 
ever, the  experiments  of  Regnault  confirmed  the  conclusion 
of  Clausius  by  showing  that  it  was  practically  constant  for 
the  so-called  permanent  gases,  as  air,  oxygen,  hydrogen  and 
nitrogen. 

Iiegnault  found  the  following  results  for  air  at  constant 
pressure  (Relation  des  Experiences,  ii.,  108). 

Heat  required  to  raise  the  temperature  of  one  pound  of 
air  1°  C.  at  constant  pressure, 

Or 

between  —  30°  C.  and  -f    10°  C.       0.23771  thermal  units, 
"  0°  C.   "    -)-  100°  C.      0.23741       "          " 

"  0°  C.   "     +  200°  C.      0.23751       "          " 

which  show  that  it  is  not  strictly  uniform,  neither  is  the  law 
of  change  apparent.  There  is,  apparently,  a  minimum 
value,  but  it  is  not  safe  to  assert  that  such  is  the  fact,  much 
less  to  assign  its  place.  Other  experimenters  find  values 
differing  slightly  from  these.  The  departure  from  the  mean 
is  so  small,  we  may,  for  all  ordinary  purposes,  consider  the 
specific  heat  as  constant. 

Kegnault  also  determined  the  specific  heat  of  air  under 
different  pressures  from  1  to  12  atmospheres,  and  of  hydro- 


[49.]  THE   PERFECTNESS   OF   A   GAS.  51 

gen  from  1  to  9  atmospheres,  and  found  the  specific  heats  of 
each  to  be  sensibly  constant  within  these  respective  ranges. 

49.  The  perfectiiess  of  a  gas  may  also  be  tested 
by  comparing  its  agreement  with  the  equation  of  a  perfect 
gas.  Thus,  Regnault  found  for  atmospheric  air,  if  the  vol- 
ume be  constant,  the  following  : 


Density,  or  pressure  in  atmospheres 
at  0°C. 

Ratio  of  pressure  at  constant  volume  at  100°  C. 
to  that  at  0°  C. 

0.1444 

1.36482 

0.2294 

1.36513 

0.3501 

1.36542 

0.4930 

1.36587 

1.0000 

1.36650 

2.2084 

1.36760 

2.8213 

1.36894 

4.8100 

1.37091 

If  the  gas  were  perfect  we  would  have  for  a  constant  vol- 
ume -y,  from  equation  (2), 

^  _p^—  p,  _     R    ,r  _  T  x  _  r,  -  T, 

fr   ~  Pi  ~   ViPi     7  T! 

in  which  the  range,  T2  —  T,,  of  temperature  being  constant, 
and  equal  100°  C.  in  the  preceding  table,  and  T,  =  273.7°, 

the  ratio  of  £*  would  also  be  constant.     The  preceding  table 

shows  a  slight  increase  in  this  ratio  with  the  increase  of  the 
density  from  0.1444  to  33.3  times  that  value.  The  depart- 
ure, however,  from  uniformity  is  so  small  that,  for  ordinary 
purposes,  air  may  be  treated  as  a  perfect  gas  in  this  re- 
spect. 

When  the  pressure  was  constant,  it  was  found  that  the 
volume  increased  as  follows  : 


PERFECT   GASES. 


[49.] 


Pressure. 

Increase  of  volume  for  an  increase  of  100°  C.,  the  original  volume 
being  unity  in  each  case. 

Atmospheric  air. 

Carbonic  acid. 

Hydrogen. 

'IGQmm. 
2525 

0.36706 
0.36944 

0.37099 
0.38455 

0.36613 
0.36616 

In  these  experiments  the  increase  of  volumes  was  greater 
for  the  same  range  of  temperatures  when  the  pressure  was 
greater,  although  for  hydrogen  the  rate  was  almost  exactly 
constant.  If  the  gas  were  perfect  we  would  have  for  a  con- 
stant pressure^,,  the  equation 


the  left  member  of  which  should  be  constant  if  the  gas  were 
perfect,  the  range  of  temperatures  and  the  initial  tempera- 
ture being  constant. 

In  some  other  experiments  the  same  mass  of  different 
gases  was  subjected  to  different  pressures  with  the  following 
•results,  v  being  the  volume  of  one  pound  of  the  mass : 


Density  or 

» 

Hydrogen. 

Nitrogen. 

Atmospheric  air. 

p              pv 

\ 

P 

pv 

P 

1.9975 
3.9860 
7.9457 
15.8045 

pv 

2 
4 
8 
16 

2.00081.0004 
4.0061  1.0015 
8.0339  1.0042 
16.16161.0101 

1.9995 
3.9918 
7.9641 
15.8597 

0.9992 
0.9979 
0.9955 
0.9912 

0.998782 
0.996490 
0.993212 

0.987780 

This  table  shows  that  these  three  gases  follow  nearly  the 
gaseous  law  expressed  by  the  equation^?  v  =  R  T»  that  for 
hydrogen  p  v  increases  slightly  with  increase  of  pressure, 
while  for  nitrogen  and  atmospheric  air  this  product  decreases 
with  increase  of  pressure. 


[50,  51.] 


TO   FIND   Cv. 


53 


The  following  table  gives  the  expansion  of  several  gases 
under  constant  pressure  from  0°  C.  to  100°  C.,  and  the  in- 
creased tension  for  the  same  range  of  temperatures  under 
constant  volumes,  the  initial  pressure  being  one  atmosphere, 
as  determined  by  Regnault. 


Substance. 

Increase  in  volume 
under  constant 
pressure  for 

Increase  of  pressure 
under  constant 
volume  for 

100°  C. 

1°F. 

100°  C. 

1°F. 

Hydrogen 

0.3661  0.002034 
0.36700.002039 
0.36700.002039 
0.3669,0.002038 
0.3710  0.002061 

0.3667 

0.3665 
0.3668 
0.3667 

0.3688 

0.002037 
0.002036 
0.002039 
0.002037 
0.002039 

Atmospheric  air  

Nitrogen 

Carbonic  oxide  

"         acid  .  . 

Protoxide  of  nitrogen 

Sulphurous  acid 

Cyanogen. 


0.3719!o.002066  0.3676|0.002032 
0.3903!0.002168;0.3S45  0.002136 
0.3877  0.002154  0.38290.002127 


5O.  To  find.  Ov.  The  specific  heat  of  any  substance 
at  constant  volume  has  not  been  found  to  any  degree  of  ac- 
curacy by  direct  experiment,  but  its  value  for  sensibly  per- 
fect gases  may  be  computed  from  equation  (29),  for  \ve 
have 

<7v  =  c\  -  R, 

which  is  the  required  equation. 

Regnault  found  for  air,  the  mean  value 

cn  =  0.2375  T.  U. 


(30) 


.-.  C\  =  184.77  =  0.2375  X 


778. 


Also,   equation  (3'),  R  =    53.37 
difference  =  Cv  =  131.40 ; 

.-.  cv  =  0.1689  =  131.40  -^-  778.  h 

Equation  (3')  is  here  used  because  the  determinations  were 
made  with  the  air  thermometer. 
51.  Relative  specific  heats.     Since  both  specific 


64  PERFECT  GASES.  [52.] 

heats  are  constant  for  perfect  gases,  their  ratio  will  be  con- 
stant, which  we  will  represent  by  y  ;  then 

' 


For  air  we  have 


This  ratio  was  originally  found  by  means  of  the  velocity 
of  sound  in  the  gas,  in  a  manner  soon  to  be  explained,  Art.  60, 
From  equations  (30)  and  (31)  we  find 

C9  =      r      B  =      Y     .&Lv0  =  Dv0,         (32) 

r-i         y-i    r0 

where  D  is  a  constant  for  sensibly  perfect  gases  ;  hence,  for 
another  gas  we  have 


that  is,  the  specific  heats  of  two  sensibly  perfect  gases  are  di- 
rectly as  their  specific  volumes. 

But  the  specific  volumes  are  inversely  as  the  specific 
weights,  or  densities,  of  the  gas,  or 

V°=w=  id' 
hence, 


that  is,  the  specific  heats  of  two  perfect  gases  are  inversely 
as  their  densities. 

52.  Let  the  temperature  be  constant  during 
expansion,  find  the  heat  absorbed. 
For  this  condition 

dr  =  o 


1 53. ' 


LET  THE   VOLUME   BE   CONSTANT. 


55 


in  the  first  and  second  of  equations  (JB\  and  from  the  first 
we  have 

H  = 


d  v, 


(35) 


which  may  be  integrated  if  p  be  a  known  function  of  v. 
The  equation  of  the  path  of  the  fluid  will  be  equation  (2), 
making  T  =  T^ 

PV  =  £TV 

the  value  of  p  from  which  substituted  in  the  preceding 
equation  gives 

^  —  =  R  TI  log  —  .  (36) 


The  first  member  of  this  equation  may 
be  represented  by  the  area  (p^AB(p» 
Fig.  19,  and  the  last  member  by  vt  A 
B  v»  which  is  the  external  work  done 
during  the  expansion  ;  hence,  in  a  per- 
fect gas  the  external  work  done  during 
an  isothermal  expansion  equals  the  heat 
absorbed — a  necessary  result,  there  be-  FIG  jg 

ing  no  internal  work. 

Since  the  area  A  B  b  is  common,  it  follows  that,  for  a 
perfect  gas, 

(p,  I  B  (p,  =  v,  A  b  v,. 


53.  Let  the  volume  be  constant. 

v  =  0  and  v  =  VH  and  equations  (B}  give 


But  the  equation  of  the  gas  gives, 

v,  p,  =  R  r,,        v,  p,  =  B  r» 
and  the  condition  of  the  problem  gives  vl  =  v9  ; 


Then  will 


(37) 


=  (<7P  -  R)  (T,  - 


56  PERFECT  GASES.  [54.] 

which  placed  equal  to  equation  (37)  gives 

as  before  found  in  equation  (29). 

54.  Let  the  pressure  be  constant.    Significance 
of  Ry  equation  (29).     Let  the  heat  absorbed  be 

d  H  =  Cvdr,  (38) 

and  equation  (B],  becomes 

and  since  p  is  constant  during  the  absorption  of  heat,  as  indi- 
cated by  the  condition  in  equation  (38),  we  have  by  integrating 
the  last  equation  between  the  limits  r  and  r  +  1  for  temper- 
ature, and  vt  and  vt  for  volumes,  observing  that  -y,  —  -y,  will 
be  the  horizontal  distance  between  the  isothermals  T  and 
r  -\-  I  at  the  upper  extremity  of  the  ordinate  p,  we  have 

that  is,  the  value  of  R  is  the  energy 
\     \  exerted    by  one  pound  of  the  gas 

A\ — \c  in  expanding   at  constant  pressure 


while  the  temperature  increases  one 
degree. 

In  Fig.  20,  if  the  isothermals  through 
A   and  B  respectively  differ  by  one 
FIGSO  degree,  A  B  being  horizontal,  B  C 

vertical,  we  have 

Op  =  m,  A  B  ra3,  Cv  =  mt  C  B  ra,, 

and  by  the  second  law, 

-y,  A  C  v,  =  ml  A  C  mt  • 
.  '  .  <7P  —  Ov  =  ml  A  B  m,  —  m,  CB  m,  =  m,AB  Cmt 


which  is  the  external  work  done  during  the  expansion  at 


[55.J 


LET   THE   PATH   BE  ARBITRARY. 


57 


constant  pressure  from  the  state  A  on  one  isothermal  to  the 
state  B  on  the  isothermal  one  degree  higher,  as  stated  above. 

For  air  this   becomes   53.21   foot-pounds   (Eq.  (3) ),    or 
53.21  -r-  778  =  0.069   of    a   thermal 
unit. 

55.  Let  the  path  he  arhitra- 
ry.      Then  will  the  first  of  equations 


The  second  term  is,  Fig.  21, 


FIG.  21. 


p  d  v  =  vl  A  B  vn 


and  may  be  separated  into  two  parts.     Through  A  draw  the 
isothermal  A  Cy  and  the  adiabatios  A  m,  C  niT  ,  B  mz  ;  then 

v1  A  C  v^  =  m  A  C  mr  ; 


/.    /  p  d  v  =  m  A  C  m,T  -f-  A 


23  C. 


But 


II  —  m  A  B  m2  ; 
v  =  m  AS  m,  -  in  A  C  mT  -  A  B  C 


=  mr  G  B  m,  =  Cv  (ra  -  r,),  (Eq.  (37))  ; 

that  is,  to  find  the  increased  energy  of  the  substance  in 
passing  from  state  A  to  state  B  due  to  the  absorption  of 
heat,  through  the  initial  state  A  of  the  substance  represented 
on  a  diagram  of  energy  pass  an  isothermal,  and  note  the 
point  C  where  it  intersects  the  ordinate  to  the  second  state, 
then  will  the  area  between  G  JB  and  two  adiabatics  drawn 
respectively  through  C  and  £  indefinitely  extended  in  the 
direction  of  increased  volume,  represent  the  increased 
energy  of  the  substance. 

Let  the  isothermal  A  C  be  prolonged  to  an  intersection 


PERFECT   GASES. 


[55.] 


with  B  <pa,  at  D,  Fig.  22,  then  according  to  the  second  law 
the  indefinitely  extended  area  q>3  CD  <p2  will  equal  v,  CDv', 

to  which  adding  the  area  BCD,  we 

have 


FIG.    22. 


by  which  means  the  increased  energy 
of  the  substance  in  the  state  B  over 
that  in  state  A  may  be  represented  by 
the  finite  area  v,  B  D  v'.  Thus  in 
working  from  -y,  to  vt  along  the  path 
A  B,  Fig.  23,  the  external  work  vtAB 
vy  will  have  been  done,  and  the  energy  of  the  substance 
will  have  been  increased  by  the  heat  absorbed  an  amount 
represented  by  the  area  v^  B  D  v'.  This  mode  of  repre- 
sentation is  due  to  M.  Cazin. 

If  a  piston  were  driven  by  the  expansion  of  a  fluid  with- 
out absorbing  or  emitting  heat,  it  would 
do  the  work  v,  A  E  v»  Fig.  22,  where 
A  E  is  an  adiabatic  ;  but  if  the  heat 
of  the  expanding  fluid  be  maintained 
constant,  it  will  do  the  work  vl  A  C  vn 
where  A  Cis  an  isothermal.  In  the 
latter  case,  the  heat  absorbed,  fptA  C 
<p3,  according  to  the  second  law,  equals 
the  entire  work  done,  vt  A  C  vt ;  but 
the  work  done  due  to  the  heat  absorbed  exceeds  that  done 
by  adiabatic  expansion  by  the  area  E  A  C. 


V,  *a    V 

FIG.    23. 


EXERCISES. 


1.  Deduce  equation  (37)  from  the  second  of  equations  {B\ 
employing  any  other  equation  necessary. 

2.  How   many  foot-pounds  of    heat  must  be   absorbed 
by   2   pounds  of  air  in   expanding  to   double    its   initial 
volume   at  the  constant  temperature   of   100°  F.?     How 


[55.]  LET   THE   PATH   BE   ARBITRARY.  59 

many  degrees  F.  would  it  raise  the  temperature  of  20 
pounds  of  water  ?  Here,  TI  =  460.66  -j-  100  =  560.66,  and, 
equation  (36), 

#  =  2  X  53.21   X  560.66  X  (2.303  X  0.301030) 
=  41370  ft.-lbs. 


3.  How  many  B.  T.  U.  of  work  must  be  expended  in 
compressing  3  pounds  of  air  at  the  constant  temperature  of 
15°  C.  to  one  fourth  its  initial  volume  ? 

4.  By  means  of  equations  (3),  (29)  and  (31)  reduce  equa- 
tions (B}  to  the  following  :  —  • 


d  H=  Op  fo. 


Tll  dr       0.288T  r  — 


dff=Cv  fl.406  d  r  -  0.406    r 

L  P  _\ 

5.  The  specific  heat  of  hydrogen   at   constant   pressure 
being  3.4090  find  the  specific   heat   at    constant   volume. 
Finl  the  ratio  of  the  specific  heat  at  constant  pressure  to 
that  at  constant  volume. 

6.  The  specific  heat  of  oxygen  at  constant  volume  having 
been  found  to  be  0.1551  find  the  specific  heat  at  constant 
pressure  ;  the  ratio  of  the  two  ;  and  their  values  in  foot- 
pounds. 

7.  Having 

C,  -  Cv  =  R, 
and 

CP  =  y  =  1.406, 

L^v 

to  find  (7P  and  (7V  in  terms  of  R. 

Ans.  Cf  =  3.463  R  =  —?—  R. 
Y  ~  1 

C\  =  2.463  B  =  _  -  _  R, 

r  —  i 


60  PERFECT   GASES.  [55.J 

8.  Find  the  value  of  D  in  equation  (32),  and  the  value  of 
Cv  in  terms  of  v0. 

9.  The  specific   heat  of  air  being  0.2375,  and  the  weight 
of  a  cubic  foot  being  0.080728  Ibs.,  and  the  weight  of  a 
cubic  foot   of  hydrogen  being  0.005592,  find  the  specific 
heat  of  the  latter  by  equation  (3-i)  and  compare  the   result 
with  the  tabular  value. 

10.  How  many  foot-pounds  of  heat  must  be  absorbed  in 
expanding  three   kilograms   of  air  at  the  constant  temper- 
ature of  30°  C.  from  3  cubic  metres  to  5  cubic  metres  ? 

11.  If  the  equation  to  the  path  of  the  gas  be  p  =  a  v  +  ft, 
the  initial  volume  vl  =  10  cu.  ft.,  initial  pressure  2000  Ibs. 
per  square  foot,  the  terminal  v,  =  20  cu.  ft.,  p,  =  5000  Ibs. ; 
how  much   heat   must  be   absorbed  in   passing   from  the 
initial  to  the  terminal  state,  how  much  external  work  will 
be  done  and  how  much  will  the  energy  of  the  substance  be 
increased  ?     Let  the  substance  be  atmospheric  air. 

Approx.  ans.  //  =  232000  ft.-lbs. 


pdv  —    35000  " 
H  —   [  p  d  v  =  197000  "     " 

12.  How  much  heat  must  be  absorbed  by  a  perfect  gas  iu 
expanding  at  a   constant  pressure  from  vt  to  0,,  the  initial 
temperature  being  rl  ?   What  will  be  the  final  temperature  ? 

13.  How  much  heat  must  be  absorbed  by  a  perfect  gas  in 
expanding  from  the  state pl9  v,  top,,  vt,  the  equation  to  the 
path  being  p*  =  m  (v  —  6)  ?     If  gas  be  air,  p,  =  2000  Ibs. 
per  ft.,p,  =  6000,  v}  =  10  cu.  ft.,  and  v,  =  20  cu.  ft. 

Ans.  290580  ft.-lbs. 

14.  In  the  preceding  example,  the  heat  absorbed  would 
raise  the  temperature  of  how  many  pounds  of  water  through 
three  degrees  F.  ? 

15.  How  many  B.  T.  TJ.  will  be  required  to  double  the 


[56.]          LET  THE  GAS   EXPAND  WITHOUT  HEAT.  61 

volume  of  one  pound  of  air  at  constant  pressure  from  the 
temperature  of  melting  ice  ? 

16.  How  much  heat  will  be  absorbed  in  expanding  a  per- 
fect gas  to  twice  its  initial  volume,  if  the  equation  to  the 
path  be  p  v7  —  c  (a  constant)  ? 

56.  Let  the  gas  expand  witUout  transmis- 
sion of  heat.  Since  no  heat  is  absorbed  or  emitted,  we 
have,  in  equations  (£), 

dH=o-, 

.'.  Cy  d  r  =  —  p  d  v, 
Opd  T  =  v  dp. 

Dividing,  gives 

dp  _        Cp  dv  _  dv 

p    '          Cv    v    '          ^    v 

Integrating, 

7   P    7   / 

log  —  =  loq  \  — 
e  Pi  e  \  v 

where  j9,  and  vl  are  the  initial  limits,  the  other  limits  being 
general.  From  this  we  find 

7  7 

p  v  =  pl  v1  =  constant,  (40) 

which  is  the  equation  of  the  projection  of  the  line  of  no 
transmission  on  the  plane  p  v.  To  find  it  on  the  plane  f  v, 
eliminate  p  from  the  preceding  equation  by  means  of  equa- 
tion (2)  and  find 


also=^-j~3  (41) 

which  are  the  equations  to  the  adiabatics  for  perfect  gases. 
If  p»  v»  ra  be  terminal  values,  then 


i  =  (v         (JM 

Tl  W,  /  \pt/ 


PERFECT   GASES.  [57.] 


EXERCISES. 

1.  Two  cubic  feet  of  air  at  60°  F.,  and  initial  pressure  one 
atmosphere  (absolute),  is  compressed  in  a  cylinder  to  5  at- 
mospheres gauge  pressure  ;  if  there  be  no  transference  of 
heat,  required  the  terminal  temperature  and  volume,  and 
the  pounds  of  water  at  50°  F.  necessary  to  reduce  the  tem- 
perature to  65°  F. 

We  have,  omitting  0.66  in  the  temperature, 

r,  =  460  +  60  =  520°  F. 

r,  =  520  (f)1^  =  873°  F. 

v,  =  2  ftp11  =  0.559. 
IF  (65  -  50)  =  2  x  0.1689(7,  -  525)  x  0.0807  x  |i| ; 

.'.  IF  =11.22+. 

2.  If  one  cubic  foot  of  air  expands  from  a  pressure  of  4 
atmospheres  gauge  pressure  and  temperature  of  60°  F.  to 
an  absolute  pressure  of  one  atmosphere  without  transmission 
of  heat,  required  the  terminal  temperature. 

We  have 

T,  =  520  tt)iW"r  =  327° ; 
.  ' .  T,  =  327  -  460  =  -  133°  F., 

or  the  terminal  temperature  will  be  133°  below  the  zero  of  Fahrenheit's 
scale. 

57.  It  will  be  seen  from  exercise  2  that  a  low  tempera- 
ture may  be  secured  by  suddenly  expanding  a  gas  from  a 
high  pressure  and  moderate  temperature  to  a  low  pressure. 
This  principle  is  used  for  commercial  purposes,  one  form  of 
which  is  called  "  cold  storage."  A  gas — as  ammonia — is 
compressed  to  a  comparatively  high  tension,  thus  increasing 
its  temperature,  and  allowed  to  cool  while  under  high  ten- 
sion, after  which  it  is  expanded  to  a  low  tension,  thus  pro- 
ducing a  low  temperature  of  the  fluid.  A  liquid  whose 


[58.]  AN   AIK-COMPRESSOR.  63 

freezing  point  is  lower  than  that  of  water — technically  called 
"  brine  "—is  made  to  circulate  in  pipes  through  the  cool  fluid, 
thus  reducing  its  temperature,  after  which  it  passes  to  com- 
paratively air-tight  compartments  containing  the  articles  to 
be  preserved — such  as  meat,  vegetables,  fruit,  eggs,  &c.  Any 
desired  temperature  may  be  thus  maintained  for  any  length 
of  time  at  all  seasons  of  the  year.  Meat  is  thus  stored  and 
kept  frozen  for  months. 

Ice  machines  are  constructed  on  the  same  principle,  by 
means  of  which  ice  may  be  manufactured  during  hot 
weather. 

58.  Ail  air-compressor  is  a  machine,  or  engine,  for 
compressing  a  gas,  as  atmospheric  air,  to  a  higher  tension. 
Air  thus  compressed  is  useful  for  many  purposes — especially 
for  driving  engines  in  the  place  of  steam,  when  the  power 
is  to  be  transmitted  a  considerable  distance,  and  especially 
underground.  It  does  not  condense  like  steam.  If  the  heat 
which  is  generated  in  the  act  of  compression  could  be  re- 
tained until  the  air  is  used  in  the  motor,  it  would  be 
useful,  but  a  large  portion  of  it  generally  escapes  through 
the  walls  of  the  conducting  pipes  and  storage  reservoirs, 
and  hence  is  energy  lost.  To  avoid  this  loss  efforts  are 
made,  in  the  use  of  the  best  compressors,  to  prevent  as 
much  as  possible  the  rise  of  temperature  during  the  pro- 
cess of  compression,  by  injecting  water  into  the  cylinder. 
The  water  should  enter  the  cylinder  in  the  form  of  a  fine 
spray,  the  elements  widely  diverging,  so  as  to  fill,  as  nearly 
as  possible,  the  entire  cross-section  of  the  cylinder  with  a 
mist. 

If  compressed  air  escaping  from  a  vessel  suddenly  ex- 
pands, its  temperature  may  be  reduced  to  such  an  extent  as 
to  freeze  the  water  in  it  and  choke  the  exhaust.  This  an- 
noyance is  reduced,  and  sometimes  prevented,  in  the  case  of 
motors,  by  causing  warm  air  to  circulate  about  the  exhaust, 
or  by  gradually  increasing  the  section  of  the  outlet.  Eefrig- 


64  PERFECT  GASES.  1.58. j 

eration  and  the  principles  of  the  air  compressor  are  discussed 
on  pages  301-306  and  in  Chapter  Y. 
EXERCISES. 

1.  An  air-compressor  whose  cross-section  is  2  square  feet 
and  stroke  3  feet  takes  in  air  at  a  tension  of  14:  pounds  per 
square  inch  and  temperature  60°  F.,  and  compresses  it  to  60 
Ibs.  gauge  pressure  without  transmission  of  heat ;  required 
the  final  temperature  and  the  pounds  of  water  at   50°  F. 
necessary  to  reduce  the  temperature  of  the  air  to  55°  F. ;  to 
70°  F. ;  to  90°  F. 

2.  In  the  preceding  exercise,  if  the  air  at  60  Ibs.  gauge  pres- 
sure and  70°  F.  expands  adiabatically  to  a  final  pressure  of  20 
Ibs.  gauge  pressure;  required  the  final  temperature  and  the 
pounds  of  water  that  it  would  cool  from  60°  F.  to  32°  F. 

3.  Required  the  entire  amount   of   heat-energy  in  one 
pound  of  atmospheric  air,  at  the  temperature  of  melting  ice, 
considered  as  a  perfect  gas. 

Considering  that  it  is  brought  to  its  present  temperature  by  being 
heated  from  absolute  zero  at  a  constant  volume,  we  have,  in  the  first  of 
equations  (B), 

d  v  =  o\ 


H=C,     l°dT  = 


=  C,   TO  («) 


=  131.40  X  492.66  =  64735  ft.-lbs.  (6) 

We  may  also  consider  that  the  entire  heat- 
energy  has  been  transmuted  into  external  work 
v  v  by  an  adiabatic  expansion,  A  H,  Fig.  24,  in 

which  case  we  have,  from  the  first  of  equations 
PIG.  24.  (5)> 

d  H  =  0  =  Or  d  T+  p  d  v. 
But  from  equation  (40), 


•.-  a./^r  = 


y    -y 

v0  v  =  a  v,  Fig.  24 ; 


158.] 


AN   AIR-COMPRESSOR. 


6V  TO  = 


7-1        7-1 
.  •  .  7  =  1.405  very  nearly. 

If  7  =  1.406,  as  previously  found,  then 


=  64735; 


=  64566, 

7-  1 

which  is  about  -^  of  this  value  less  than  the  preceding. 

If  the  terminal  temperature  be  the  zero  of  the  air  thermometer,  then 
the  value  of  Hia.  equation  (a)  would  be 

H  =  131.40  X  491.13  =  64534, 
which  agrees  very  nearly  with  the  preceding  value,  where  7  =  1.406. 

4.  Eequired  the  height  to  which  a    ball  weighing  one 
pound  could  be  projected  upward  in  a  vacuum  by  the  heat- 
energy  in  one  pound  of  air  under  the  pressure  of  one  at- 
mosphere at  the  temperature  of    melting   ice.      (Use  the 
value  in  (b)  of  exercise  3.)     How  many  times  the  height  of 
a  homogeneous  atmosphere  ? 

5.  Required  the  entire  heat-energy  in  one  pound  of  hy- 
drogen at  sea  level  at  the  temperature  of  melting  ice. 

6.  Required  the  heat-energy  in  two  pounds  of  air  under 
the   pressure   of    one    atmosphere   at  the  temperature   of 
100°  F. 

7.  If  a  gas  be  forced  into  or  out  of  a  receiver  of  constant 
volume,  without  transmission  of  heat,  can  equations  (42) 
be  transformed  so  as  to  give  the  rela- 

tion between  weights,  temperatures  and 
pressures  ? 

8.  For  a  perfect  gas,  verify  the  fact 
that  the  external  work  vl  A  B  v»  Fig. 
25,  equals  (plAB(p»AB  being  an  iso- 
thermal, A  9?,  and  B  q>t  adiabatics,  by 
finding  the  area  tp1  ~b  B  <pv 


The  equation  of  A  g>i  will  be 


Y          v 
Pi  vl  =  pv , 


PERFECT   GASES. 


of  B  (p3,  ptvt=pv; 


p=ec  =  PLAFig.25, 


from  the  former, 


from  latter, 


and 

area  g>ibBcp,= 


i-»dr* 

'J*.     vY 


y-1 

y    I-Y 
Also,  /•"«      »  rf  v       ^i  »i  —  jQi  v.  v» 

area  viAbvt=l     p,  v,  -—  — y~^~l ' 

•'F,  v 

and  since  ^4  5  is  an  isothermal,  we  have  pi  PI  =  p*  v,  •  hence 

area  <pt  b  B  <p9  =  area  v,  A  b  vt. 
Adding  A  B  b  A  to  both,  gives 

vt  A  Bvt  =  g)i  A  B  <p,. 

9.  In  a  perfect  gas,  show  that  if  the 
states  A  and  B,  Fig.  26,  are  on  the 
same  isothermal,  the  intrinsic  energies 
are  the  same  at  those  states,  and  hence 
independent  of  the  path  of  the  fluid 
between  those  states. 


0     V,        V»  This  follows  directly  from  the  analysis  of 

FIG.  26.  exercise  3  ;  for  it  is  there  shown  that  the  intrin- 

sic energy  at  A  will  be 


and  hence  at  B  ^b- 

7-1' 


[58.]  AN   AIR-COMPRESSOR. 

but  since  A  and  B  are  on  the  same  isothermal,  we  have 


67 


and,  therefore,  the  two  preceding  expressions  are  equal,  independently 
of  the  path  between  the  states  A  and  B. 

10.  In  a  perfect  gas,  verify  the  principle  :  The  elementary 
areas  between   consecutive  isothermals 
and    a  pair  of  adiabatics  are  equal,  by 
using  the  equations  to  the  adiabatics. 

Let  A  B,  Fig.  27,  be  an  isothermal—  the  top- 
most one  —  y  z  any  other,  both  limited  by  the 
pair  of  adiabatics  A  <pi  and  B  <p?.  Let  isother- 
mals be  drawn  under  A  B  and  y  e  respectively, 
so  that  the  difference  in  temperatures  will  be 
d  T  in  each  ;  the  length  of  the  one  at  A  B 
measured  on  the  axis  of  abscissas  being  v3  — 
to  prove  that 


FIG.  27. 

,  and  of  yz,  vi  —  vs,weare 


r 

= 
Jv, 


dp  dv. 


Let  the  equations  to  the  lines  be 

p  v  =  a  —  R  TI  for  A  B, 
p  v  =  d  =  R  T 3  for  y  z, 

pv  =  b  for  A  <pi, 
p  v  =  e  for  B  <pi. 


(D 
(2) 

(3) 


To  find  the  abscissa,  vlt  of  the  point  of  intersection  of  A  B  and  A  <p\, 
make  p  common  to  equations  (1)  and  (3),  eliminating  which  gives 

1 


y-l 


Similarly,  from  (1)  and  (4), 


1 

y  —  l' 


70  PERFECT   GASES.  [58.] 


0  Vi  =  v\  —  (  —  )  Y  ~  *,  and  similarly  for  the  other  co-ordinates.  We  have 

vlAlB1T3=    I  pdv  =  c3    I     3<L*=  _L_  (<-,  -  (j,), 

/  /        fl?       y  —  1 

%/  i/ci 

and  the  area  «»  At  5a  u«  reduces  to  the  same  value  ;  hence,  in  working 
around  Carnot's  cycle  the  work  done  during  expansion  without  trans- 
mission of  heat  equals  that  done  during  compression  also  without  trans- 
mission of  heat,  and  they  cancel  each  other.  This  should  have  been  an- 
ticipated, since  all  the  work  done  is  by  a  transmutation  of  heat. 


12.  A  cylindrical  vessel  the  area  of  whose  base  is  one 
square  foot  contains  2  cubic  feet  of  air  at  60°  F.  when  com- 
pressed by  a  frictionless  piston  of  2000  pounds  resting  upon 
it  ;  required  the  volume  and  temperature  of  the  air  if  the 
vessel  be  inverted,  there  being  no  transmission  of  air  or 
heat. 

13.  A  cylindrical  vessel  the  area   of  whose   base   is   a 
square  feet  and  height  b  feet  is  filled  with  air  at  60°  F.  and 
pressure  of  one   atmosphere;   a  frictionless   piston   whose 
weight  is  w  is  placed  at  the  upper  end  and  dropped  into  the 
cylinder  ;  if  there  be  no  escape  of  air  nor  of  heat  how  much 
will  the  air  be  compressed,  and  what  will  be  its  temperature 
at  the  instant  of  greatest  compression? 

Let  *  =  the  height  of  the  volume  of  air  when  the  piston  has  descended 

to  its  lowest  point  ; 

p  =  the  pressure  of  the  atmosphere  on  a  square  foot  ; 
then  will  the  work  done  during  the  descent  of  the  piston  be 

(p  a  +  tr)  (b  -  x}  ; 
and  the  opposing  work  done  by  the  air  will  be 


from  which  the  value  of  .r  may  be  found  when  numbers  are  substituted 


[59.]  VELOCITY   OF   A   WAVE.  71 

for  the  other  letters,  and  therefore  b  —  x  becomes  known.    The  adiabatic 
relation  also  gives 


14.  If  the  heat  absorbed  varies  as  rn  find  the  path  of  the 
fluid. 
Here  H  =  a  rn  ; 

.-.  dH=  anr"-1  dr, 
and  the  first  and  second  of  equations  (B)  give 

anrn~ldT  =  C-v  dr  -\-  p  dv, 
an  rn  ~    dr  =.  Cpdr  —  v  dp; 
in  which  substitute 

9=*!^=*!, 

v  p 

and  dividing  through  by  r,  thus  separating  the  variables,  and  integrating, 


Eliminate  r  by  means  of  equation  (2)  and  find  the  equation  in  terms  of  p 
and  v.     Equation  (2)  may  be  deduced  from  the  last  equation  here  given. 

15.  If  the  heat  absorbed  varies  as  v*  find  the  equation  to 
the  path  of  the  fluid. 

16.  If  the  heat  absorbed  varies  as  j?n  find  the  path  of  the 
fluid. 

17.  If  H  •=.  a  vn  -|"  ^  P*  find  the  equation  to  the  path. 

APPLICATIONS. 

59.  Velocity  of  a  wave  in  an  elastic  medium,* 

This  article  is  a  digression  for  the  purpose  of  establishing  a  formula  from 

*  The  general  problem  of  wave  propagation  has  received  the  attention 
of  several  of  the  most  eminent  mathematicians  since  the  days  of  Newton, 
and  many  problems  have  been  solved  in  a  satisfactory  manner.  The 
simple  method  of  Newton,  Principia,  Prob's  XLIII.-L.,  B.  II.,  has  not 
been  excelled,  and  the  definite  theoretical  result  obtained  is  quoted  to 
the  present  day,  although  the  effect  of  heat  upon  the  velocity  of  sound 
was  net  then  known.  La  Place,  in  the  Mecamque  Celeste,  tomes  II.  and 
V.  ,  has  treated  of  the  oscillations  of  the  sea  and  atmosphere  ;  Lagrange, 


72  PERFECT   GASES.  [59.] 

which  7  may  be  deduced.  Assume  that  the  medium  is  confined  in  a 
prismatic  tube  of  section  unity,  E  the  coefficient  of  elasticity  for  com- 
pression,^ a  force  which  will  produce  a  compression  d  y  in  a  length  d  x, 
then  from  definition  we  have 


The  lamina  d  x  will  be  urged  forward  —  or  backward  —  by  the  differ- 
ence of  the  elastic  forces  on  opposite  sides  of  it,  and  as  the  quantities  are 
infinitesimal,  this  difference  will  be  dp^  •  or 


Let  D  be  the  density  of  the  lamina,  then  its  mass  will  be  M  =  D  d  x, 
and  we  have  from  equation  (21),  page  18,  of  Analytical  Mechanics, 


_  (43) 

4f-  JJd  **' 

which  is  a  partial  differential  equation  of  the  motion  of  any  lamina,  the 
integral  of  which  is  given  in  works  on   Differential   Equations.  One 

of  the  methods  is  as  follows  :    Let  E  -^-  D  =  a3,  and  adding  a    ^y 

dx  dt 
to  both  members,  we  have 


>dt 

Let 


jj' (if +•£)-£*(& 


-- 
dt          dx' 


in  the  Mecanique  Analytiquz,  tome  II.,  has  discussed  the  problem,  of  the 
movement  of  a  heavy  liquid  in  a  very  long  canal ;  M.  Navier  published 
a  Memoire  on  the  flow  of  elastic  fluids  in  pipes,  in  the  Acddemie  des 
Sciences,  tome  IX. ;  and  M.  Poisson  wrote  several  Memoires  on  the  propa- 
gation of  wave  movements  in  an  elastic  medium,  and  the  theory  of 
sound,  for  which  see  Journal  de  I'Ecole  Poly  technique,  14th  chapter,  and 
of  the  Academie  des  Sciences,  tomes  II.  and  X.  These  eminent  mathe- 
maticians established  the  basis  of  the  analysis  for  the  solution  of  the 
problem.  More  recently  we  have  M.  Lame's  Lecons  »ur  I'Elasticite,  des 
Corps  solves,  and  Lord  Rayliegh's  Treatise  on  Sound,  both  of  which  are 
works  o*  ?reat  merit. 


[59-]  VELOCITY   OF   A   WAVE.     '  73 

tben  (*I\  =  *Jf  +  a  -*J-  (44) 

\dt)        dt*^      dxdt' 

where  the  parenthesis  indicates  a  partial  differential  coefficient,  and 


__ 

dx  /        d  td  x   '       dx* 


(45) 


and  equations  (43),  (44),  (45),  or,  more  directly,  the  last  equations  on 
p.  72  give 

(*n=a(!*Z\  (46) 

\dt  /  \dx  J 

The  total  differential  of  V  =  f(x,  t)  is 


by  substituting  (46), 


and  integrating, 

^  V    I      dy  fAr>\ 

-rr+a-7— ,  (*() 


where  F  is  any  arbitrary  function. 

Similarly  subtracting  a  — — —  from  (43), 

F1  —  f(    —     f\  —  ^^          ^  ,y  /4g) 

Adding  and  subtracting  (47)  and  (48),  we  have  the  respective  equations 


But 

and  substituting  from  above,  gives 
integrating, 


=  -L  F  (x  +  a  0  d  (x  +  a  0  - ~  f(x  -  a  f>  d  (x  -  a  f) ; 
a  (t  6  (t 


+  a  0  -  <?  (x  —  a  f),  (49) 


72  PERFECT   GASES.  [59.] 

which  y  may  be  deduced.  Assume  that  the  medium  is  confined  in  a 
prismatic  tube  of  section  unity,  E  the  coefficient  of  elasticity  for  com- 
pression,j9ia  force  which  will  produce  a  compression  d  y  in  a  length  d  x, 
then  from  definition  we  have 

P>  =  EdJL. 
^  dx 

The  lamina  d  x  will  be  urged  forward  —  or  backward—  by  the  differ- 
ence of  the  elastic  forces  on  opposite  sides  of  it,  and  as  the  quantities  are 
infinitesimal,  this  difference  will  be  dp^  •  or 

*  =  **=*  IT* 

Let  D  be  the  density  of  the  lamina,  then  its  mass  will  be  M  —  D  d  x, 
and  we  have  from  equation  (21),  page  18,  of  Analytical  Mechanics, 


which  is  a  partial  differential  equation  of  the  motion  of  any  lamina,  the 
integral  of  which  is  given  in  works  on   Differential   Equations.  One 

of  the  methods  is  as  follows  :    Let  E  -L.  D  =  a*    and  adding  a        $ 

dxdt 
to  both  members,  we  have 

Let 


in  the  Mecanique  Analytiqua,  tome  II.,  has  discussed  the  problem,  of  the 
movement  of  a  heavy  liquid  in  a  very  long  canal ;  M.  Navier  published 
a  Memoire  on  the  flow  of  elastic  fluids  in  pipes,  in  the  Academie  des 
Sciences,  tome  IX. ;  and  M.  Poisson  wrote  several  Memoires  on  the  propa- 
gation of  wave  movements  in  an  elastic  medium,  and  the  theory  of 
sound,  for  which  see  Journal  de  I'Ecole  Poly  technique,  14th  chapter,  and 
of  the  Academie  des  Sciences,  tomes  II.  and  X.  These  eminent  mathe- 
maticians established  the  basis  of  the  analysis  for  the  solution  of  the 
problem.  More  recently  we  have  M.  Lame's  Lemons  snr  I'Elattidte  des 
Corps  solves,  and  Lord  Rayliegh's  Treatise  on  Sound,  both  of  which  are 
works  G*  ^eat  merit. 


[59.]  VELOCITY   OF   A   WAVE.     '  73 

then  (*-I\  =  *»  +  a  -^J-  (44) 

\dt )       d£*          dxdt  ' 

where  the  parenthesis  indicates  a  partial  differential  coefficient,  and 


a  a-J(,  (45) 

and  equations  (43),  (44),  (45),  or,  more  directly,  the  last  equations  on 
p.  72  give 

^)=a(|Z\  (46) 

The  total  differential  of.V  =  f(x,  t)  is 

by  substituting  (46), 

—  fU-i  (d  x  +  a  d  t) 


and  integrating, 

where  Fis  any  arbitrary  function. 

Similarly  subtracting  a  -    '  y    from  (43), 
dxdt 

r  =  f(x-af)  =  ^L-a^.  (48) 

Adding  and  subtracting  (47)  and  (48),  we  have  the  respective  equations 
d-jL  =  $  F(x  +  a  0  +  i  f(x  -  a  t), 


But 


and  substituting  from  above,  gives 

d  y  =  2^  F  (a?  +  a  t)  d  (x  +  a  t)  -  ^  f(x-at)d(x-af); 

integrating, 

y  =  V  (-c  +  a  t)  ~  <P  (x  -  a  t),  (49) 


74  PERFECT  GASES.  [59.] 

•where  V  and  <p  are  any  arbitrary  functions  whateve'r.  Their  character 
and  initial  values  must  be  determined  from  the  conditions  of  the  prob- 
lem. The  equation  represents  a  wave  both  from  and  toward  the  origin. 
If  the  wave  be  from  the  origin  only,  the  q>  function  may  be  suppressed, 
and  we  have 

y  =  1>(x  +  at),  (50) 

and  differentiating, 


*l\ 

dx) 


V  (*  +  a,  0, 


which  is  the  rate  of  dilation  (the  expansion  or  contraction  of  a  prism  of 
the  air),  and 


which  is  the  velocity  of  a  particle,  and  dividing  the  latter  by  the  former, 

57  =  *  <«> 

which  is  the  velocity  of  the  wave  ;  hence, 


u  =  a  =  <y  -jy 

which  is  Newton's  formula  (Principia,  ii.,  §  8). 

The  elasticity  of  air  equals  its  tension  ;  hence,  if  p  be  the  pressure  per 
square  foot,  w  the  weight  of  a  cubic  foot,  and  H  the  height  of  a  homo- 
geneous atmosphere,  then 

u  =  yg-2-  =  4/tflT;  (53) 

hence  the  velocity  of  sound  would  equal  the  velocity  of  a  body  falling 
through  a  height  equal  to  one  half  the  height  of  a  uniform  atmosphere  of 
that  substance. 

This  principle  is  applicable  also  to  the  vibration  of  elastic  cords,  and  it 
is  found  that 

The  velocity  of  vibration  of  an  elastic  cord  equals  the  velocity  of  a  body 
falling  freely  through  a  height  equal  to  half  the  length  of  the  same  cord 
whose  weight  would  equal  the  tension. 

Similarly,  for  long  waves,  or  waves  on  water  whose  depth  is  small 
compared  with  the  wave-length, 

The  velocity  equals,  anproximately ,  the  velocity  of  a  lody  falling  freely 
through  a  height  equal  to  half  the  depth  of  the  sea.  (Ency.  Britannica.) 

It  has  been  assumed  that  E  and  D  remain  constant  in  wave  motion  ; 
but  it  has  long  been  known  that  the  results  given  by  equation  (53)  for 
gases  do  not  agree  with  those  'ound  by  experiment,  and  La  Place  showed 
that  the  elasticity  was  increased  by  the  action  of  the  wave  due  to  com- 


[60,  61.]  TO   FIND   THE  VALUE   OF   y.  75 

pression.    It  is  necessary,  therefore,  to  consider  equation  (52)  correct 
only  for  ultimate  values  ;  or 

«=  */§  =  /§ 

Since 


p  oc 


(55) 

6O.  To  find  the  value  of  7,  we  have  from  equa- 
tion (55) 

r  =  -fi>«';  (56) 

by  means  of  which  y  may  be  found  when  the  velocity  of 
sound  in  a  gas  of  given  weight  and  tension  are  known.  We 
have 

w        1  1      T0 

P         p  V        po  Vo    T 


which  reduces  the  determination  of  y  to  that  of  the  velocity, 
u,  of  sound  in  a  gas  at  known  temperature  r;g,p0  v0,  being 
known. 

61.    The  velocity  of  sound  has  been  determined 
by  direct  experiment  with  the  following  results  : 

Velocity  per  second. 
In  dry  air  at  0°  C.  Centimetres.  Feet. 

MM.  Bravais  and  Martins 33237  1090.5 

Hr.  Moll,  Yan  Beck,  and  Kuytenbrouwer  33226  1090.1 

The  French  Academy,  1738 33200  1089.2 

"          "  "          1822 33120  1086.6 

M.  Kegnault 33070  1085.0 

M.  Le  Koux . .  .   33066  1084.9 


Mean.  ...  .    33153         1087.7 


76  PERFECT  GASES.  [61.] 

Again,  if  v  be  the  velocity  of  sound,  A  the  wave  length 
and  n  the  number  of  waves  (or  the  frequency  of  the  waves) 
in  a  second,  then 

v  =  n  A.  - 

By  this  means  it  has  been  found  for  the  velocity  of  sound 
in  dry  air  at  0°  C. 

Centimetres.  Feet. 

By  M.  Dulong  ..................  _____  33300  1092.5 

"  Hr.  Seebeck  ......................  33277  1091.8 

"  M.  Schneebeli  ....................  33206  1089.4 

"  Hr.  Wertheim  .....................  33313  1093.0 

Mean  .....................   33274         1091.6 

Hr.  Kayser  determined  very  accurately  the  wave  length 
A,  by  means  of  Professor  Kundt's  dust  figures,  correspond- 
ing to  a  given  tone  of  vibration,  or  frequency  ??,  by  which 
means  he  found  for 

Centimetres.  Feet. 

the  velocity  of  sound  in  air  at  0°  C  ......   33250         1090.9 

Mean  of  the  eleven  determinations  ......   33206         1089.4 

The  height  of  a  homogeneous  atmosphere  of  dry  air  at 
0°  C.  as  determined  by  Regnault  is  20214  feet;  hence,  the 
theoretical  velocity  of  sound  in  air,  neglecting  the  effect  -of 
wave  condensation,  would  be 


v  =    /26214  X  32.2  =  918.7  ft.  per  second  ; 

/1089  4\a 

•'•  r  =  ~         =  L4061  =  1-40 


This  value  of  y  is  on  the  supposition  that  air  is  a  perfect 
gas,  and  the  error  resulting  from  this  hypothesis  is  scarcely 
appreciable,  and  certainly  cannot  affect  the  result  so  much 
as  the  errors  of  observation.  The  mean  of  such  a  large 
number  of  good  determinations  is  probably  more  reliable 
than  any  single  observation  taken  at  random.  This  was  the 
original  mode  of  finding  the  value  of  y,  and  the  only  one 
known  before  the  determination  of  Joule's  equivalent.  It 


[62,  63.  J       SPECIFIC   HEATS   OF   PERFECT   GASES.  77 

is   nearly   the   same   for   all  the  permanent  gases,  as   air, 
hydrogen,  oxygen  and  nitrogen. 

62.  To  find  the  specific  heats  of  the  sensibly 
perfect  gases.     We  found  in  equation  (32)  that 


.-.  Cv  =  B.  (60) 

For  air  we  have 


53.37  =  184.83 ; 

~r\j\j 

.-.  cp  =  184.83  -r-  778  =  0.23757. 

The  mean  of  the  values  given  by  Regnault  is  0.23751 
{Relation  desJExp.ftome  II.,  p.  101),  the  first  four  figures  of 
which  agree  with  the  results  of  our  computation.  Professor 
Rankine  was  the  first  to  make  this  computation,  in  1850, 
using,  however,  y  —  1.4,  which  gave  cv  =  0.24.  Up  to 
that  time  this  was  the  most  accurate  determination  of  the 
specific  heat  of  air ;  and  when,  soon  afterward,  the  very 
accurate  and  entirely  reliable  experiments  of  Regnault  gave 
very  nearly  the  same  result,  Rankine's  determination  was 
considered  a  crucial  test  of  the  correctness  of  the  dynamical 
theory  of  heat. 

63.  To  find  the  mechanical  equivalent  of 
heat  by  means  of  the  specific  heat  of  a  gas.  From  equa- 
tion (59)  we  have 

C»  =  Jcf  =  ^jlt; 

"  J  =  7-1"? 

Having  found  y  by  means  of  equation  (58)  without  a 
knowledge  of  the  value  of  J,  and  R  and  cp  having  been 
found  by  Regnault,  we  have,  by  substitution, 

1406        53.37 
J  =  "406    X  02375  =  7'8  foot-Po™ds 


78  PERFECT   GASES.  [63.] 

on  the  scale  of  the  air  thermometer.  If  R  =  53.21  we 
find  J  —  776  for  the  probable  mechanical  equivalent  on 
the  absolute  scale.  The  former  equals  426.8  kilogram  - 
metres.  This  was  substantially  the  method  originally  used 
by  Mayer  of  Germany,  by  which  means,  in  1842,  he  found 
365  kilogram-metres,  and  Holtzman,  in  1845,  found  374. 
The  fact  that  Dr.  Mayer  assumed  air  to  be  a  perfect  gas, 
and  made  no  attempt  to  prove  the  correctness  of  the 
assumption,  added  to  the  fact  that  the  value  he  obtained 
was  scarcely  a  rough  approximation,  has,  in  the  eyes  of 
some  historians,  deprived  him  of  the  honor  of  being  the 
first  to  determine  this  important  constant.  Joule  justly  has 
the  credit  of  first  determining  it  accurately.  Mayer  did  not 
work  out  the  equations  as  above,  but  solved  the  problem  in 
the  most  elementary  manner,  the  process  for  which  is 
worthy  of  special  study.  Thus,  he  considered  that  it  re- 
quired a  certain  amount  of  heat  to  increase  the  temperature 
of  a  given  amount  of  gas  at  constant  volume  —  all  of  which 
simply  made  the  gas  hotter,  and  if  the  gas  expanded  against 
an  external  resistance  it  required  more  heat  in  order  to 
maintain  the  higher  temperature  —  thus  reviving  the  idea 
of  Rumford  and  Davy.  Let  one  pound  of  air  under  the 
pressure  of  one  atmosphere,  p  pounds  per  square  foot, 
occupy  the  volume  v.  The  increase  of  volume  for  one 
degree  of  temperature,  p  being  constant,  will  be  v  -r-  r,  and 

hence  the  external  work  done  will  be  —  —  =  R  (equation 

(2)).  The  heat  absorbed  at  constant  volume  will  be  cy, 
and,  at  constant  pressure,  cp,  and  the  difference  of  these 
amounts  of  heat  does  the  work  fi,  provided  no  internal 
work  is  done  ;  hence, 


J=      R  ____  R 

~  cv-  cv~  (y  —  1 
as  before. 


[64,  65.]     OTHER   METHODS   OF   DETERMINING 


79 


64.  The  constants  J,  7?,  y,  are  so  related  as  to  serve  as 
mutual  checks  upon  each  other,  but  this  relation  does  not 
determine  the  exact  values  of  any  one  of  them.     When  de- 
termined directly  they   are   subject  to   small   errors,    due 
chiefly  to  errors  ~of  observation,  but  the  results  are  believed 
to  be  correct  within  £  of   one  per  cent,  and  in  some  cases 
the  error  is  probably  much  less. 

65.  Other  methods  of  determining  y.      This 
constant  has  been  found  by  the  principle  of  adiabatic  ex- 
pansion.    Thus,  equations  (41)  give 

log  p  —  logp, 

log  T:  —  log  rl 
~  log  v1  —  log  v      • 


i  —  log-c-\-  log  rl 

To  secure  data  for  use  in  these  equations,  MM.  Clement 
and  Desormes  used  a  20-litre  glass  globe  closed  by  a  stop- 
cock A,  Fig.  29,  and  connected  with 
a  vertical  glass  tube  B,  dipped  into 
water,  which  acted  as  a  manometer. 
By  means  of  an  air-pump  attached  at 
D  a  partial  vacuum  was  produced  in 
the  vessel,  after  which,  by  opening  the 
cock  at  A  a  very  short  time,  air  would 
rush  in  and  produce  the  pressure  of 
the  external  atmosphere,  and  by  com- 
pressing that  already  in  the  vessel,  raise  the  temperature ; 
and  after  the  cock  was  closed  it  cooled  to  that  of  the  sur- 
rounding temperature,  and  the  pressure  diminished.  In  the 
preceding  equations  let  p  =  p0  =  the  atmospheric  pressure, 
pl  the  initial  and  p  its  final  pressure  ;  then  the  temperature 
being  the  same  at  the  beginning  and  end  of  the  experiment, 


FIG.  29. 


80  PERFECT  GASES.  [65.] 

we  have 

v'  _  £- 

v  ~  pC 
or 

log  v1  —  log  v  =  log  p  —  log  pl  ; 


l°ff  p  — 
It  was  found  in  one  of  the  experiments  that 

p0  =  1.0136,^,  =  0.9953,  p  =  1.0088; 
.  •  .  y  =  1.3524.* 

MM.  Gay  Lussac  and  Wilter  modified  this  experiment  by 
forcing  air  into  the  vessel  and  allowing  it  to  escape  adiabati- 
cally  until  the  pressure  in  the  vessel  equalled  that  of  the 
external  air.  They  found 

j)0  =  1.0096,^,  =  1.0314,  j>  =  1.0155; 
.  -.  Y  =  1.3745.| 

In  this  manner  M.  Ilirn  found  1.3845,  M.  Dupre, 
1.399,  Hr.  Weisbach,  1.4025,  M.  Masson,  1.419  for  air.  For 
carbon  dioxide  M.  Masson  found  1.30. 

The  discrepancy  in  the  results  arises  chiefly  from  the  fact 
that  the  changes  in  pressure  are  not  adiabatic,  but  the 
inertia  of  the  inflowing  gas  produced  a  compression  exceed- 
ing the  normal  value,  resulting  in  a  reaction  tending  to  force 
a  portion  of  the  air  out  again,  producing  an  oscillating  effect, 
as  shown  by  M.  Gazing  who  also  found,  by  similar  means, 
the  same  value,  1.41,  for  air,  oxygen,  nitrogen,  hydrogen 
and  carbon  monoxide. 

Hr.  Kohlrausch  substituted  an  aneroid  for  the  manometer 
used  above,  because  more  sensitive  to  pressure,  and  found 
Y  =  1.296  (Poffff.  Annalen,  1869,  CXXXVL,  618).  This 

*  Jour,  de  Physique,  LXXXIX.  (1819),  428. 
t  Ann.  de  C%.  et  dt,  Phys.,  1821,  XIX.,  436. 
\Ibid.,  1862,  LXVI.,206. 


[06.  J  FLOW    OF   GASES.  81 

result  was  not  considered  good  on  account  of  the  small 
quantity  of  air  used  in  the  experiment,  although  the  method 
is  considered  an  improvement  on  the  preceding. 

Dr.  Kontgen  in  1872  made  a  series  of  experiments  with 
a  more  perfect  apparatus,  containing  a  much  larger  quan- 
tity of  the  gas,  the  mean  of  ten  good  experiments  giving 
Y  =  1.4053  for  air  (Hid.  (1873),  CXLVIIL,  580). 

The  difficulty  in  these  experiments  of  obtaining  the  ob- 
servations for  strictly  adiabatic  changes  generally  results  in 
too  small  a  value  for  this  constant. 

66.  Flow  of  gases.  The  flow  of  perfect  gases  as  af- 
fected by  the  principles  of  thermodynamics  was  investi- 
gated by  Messrs.  Joule  and  Thomson  (Proc.  Hoy.  .Soc., 
1856)  and  Weisbach  (Civilingeneur,  1856).  See  the  au- 
thor's Analytical  Mechanics,  page  389. 

Let  w  —  the  weight  of  a  unit  of  volume, 

p  =  the  pressure  at  any  point  of  the  issuing  jet, 
Y  =  the  velocity  at  the  point  where  p  is  measured ; 

then,  for  a  unit  of  section  and  distance  d  s  the  mass  moved 
will  be  w  d  s  -j-  y  and  the  work  done  \>y  d  p  will  be 

,  w  d  s 
d  p  d  s  =  £ d(Y)i 

g 

YI  —  r^  p 

2  g      J     w 

If  the  cooling  due  to  the  expansion  during  discharge 
follows  the  adiabatic  law,  then  from  equations  (41)  we  have 


82  PERFECT  GASES.  [66.J 


which  substituted  above  gives 
i 


Zl  =  £?  r*'d£  =  _L_pirl_T,-\. 

2g~  w,J  1   -  Y  -  1  w,  L        T,J  ' 

JT  2  /vjY 


(62) 


where  pl  is  the  tension  just  within  the  reservoir,  and^,,  that 
just  outside.     But  the  equation  of  a  perfect  gas  reduces  to 


p±  —  -P°  r'  —          It . 


which  in  English  units  becomes 


=  14.933  V   ^?  r4  (l  -  -')  ;  (63) 

TO  Tj/ 

and  this  for  air  becomes 

F  =  14.933  V  53.21  r,  (l  —  -*} 

i 

=  108.93  |/  TI  f  i  _  !•  V  nearly. 

i 
If  the  flow  be  into  a  vacuum,  p^  =  o  ;  .  • .  T,  =  o,  and 

F=  108.93   |/ir; 
which  at  the  temperature  of  melting  ice  becomes 

F=  108.93  i/492^6  =  2417  feet  per  second. 
Making  Ta  =  o  in  equation  (62)  and  comparing  with  equa- 
tion (56)  of  Article  60,  shows  that  the  velocity  of  discharge 
into  a  vacuum  will  be 

J     2 


7-1 
times  the  velocity  of  sound  in  the  gas  at  the  melting  point 


[67.]  THE  WEIGHT  OF  GAS. 

of  ice,  which  for  air  becomes  2.214  X  1089.4  =  2417  feet 
per  second,  which  is  less  than  half  a  mile  per  second. 
67.  The  weight  of  gas  escaping  per  second  will  be 


r  - 

in  which 

Q  =  the  volume  escaping  measured  outside  the  reservoir, 
w^  =  the  weight  of  unity  of  volume  outside  the  reservoir, 
S  =  the  section  of  the  orifice,  and 
Jc  —  the  coefficient  of  efflux. 

Equation  (64)  is  a  maximum  for 

2 


which  for  air  becomes 

A  =  0.831,        £L  =  0.527,        ^  =  1.577 
*,  Pi  v* 

The  values  of  k  as  found  by  Professor  Weisbach  are  : 
Conoidal  mouthpieces,  of  the  form  of  the  contracted  vein, 
with  effective  pressures 
of  from  0.23  to  1.1  atmospheres  .............  0.97  to  00.99 

Circular  orifices  in  thin  plates  ...............  0.55  to    0.79 

Short  cylindrical  mouthpieces  ................  0.73  to    0.84 

The  same  rounded  at  the  inner  end  ...........  0.92  to    0.93 

Conical  converging  mouthpieces,  the  angle  of 

convergence  being  7°  9'  ..................  0.90  to    0.99 

EXERCISES. 

1.  For  a  perfect  gas,  if  the  temperature  of  the  gas  at  the 
outside  of  the  orifice  equals  that  of  the  reservoir,  what  will 
be  the  velocity  of  exit  ? 


84  PKKKECT    GASES.  I67-] 

2.  What    is    the    initial   velocity    with    which  hydrogen 
will  flow  into  a  vacuum  from  a  vessel  in  which  the  tempera- 
ture is  60°  F.  ? 

3.  What  weight  of  air  will  flow  from  a  very  large  vessel 
in  one  second  in  which  the  internal  pressure  is  4  atmospheres 
and  temperature  100°  F.,  the  external,  one  atmosphere  and 
temperature  40°  F.,  flowing  through  a  short  cylindrical  tube 
|  inch  in  diameter,  the  coefficient  of  discharge  being  0.8  ? 
Consider  the  vessel  so  large  that  the  pressure  may  be  con- 
sidered constant  during  the  discharge. 

SUGGESTIONS    FOR    REVIEW. 

What  does  R  represent  ?  In  Fig.  SO,  may  A  be  taken  anywhere  on  the 
i-  isothermal  ?  Draw  several  verticals  between  two  isothermals  differing 
by  unity  and  show  what  areas  must  be  equivalent  if  Cv  be  constant. 
In  Fig.  22,  if  the  gas  be  compressed  from  state  B  to  state  A,  show  what 
changes  take  place  in  the  heat.  Do  the  principles  applicable  to  expansion 
also  hold  for  compression  ?  What  is  a  perfect  gas?  What  is  a  thermal  capac- 
ity ?  Define  the  two  more  common  specific  heats.  Can  there  be  more  than 
two  specific  heats  ?  Illustrate.  If  a  pound  of  air  occupies  10  cubic  feet,  and 
another  pound  40  cubic  feet,  both  at  the  same  temperature,  which  will 
absorb  the  more  heat  in  having  its  temperature  raised  one  degree  ?  De- 
scribe Mayer's  method  of  determining  the  mechanical  equivalent  of  heat. 
What  gas  has  the  greatest  specific  heat  ?  If  the  mechanical  equivalent 
were  the  heat  necessary  to  raise  the  temperature  of  one  pound  of  air  one 
degree,  about  what  would  be  its  numerical  value  ?  Describe  methods  of 
finding  the  value  of  y.  What  is  the  smallest  value  of  y  given  in  the  text  ? 
the  largest  ?  the  value  adopted  ?  How  was  the  last  one  determined  ? 
Will  the  greatest  wlnine  of  a  gas  escape  from  an  orifice  when  the  velocity 
of  exit  is  greatest  ?  Will  the  greatest  weigM  of  gas  escape  when  the  veloc 
Ity  is  greatest  ?  Why  not  ? 


CHAPTEE  III. 


IMPERFECT     FLUIDS. 


68.  General  discussion.  Equations  (A)  are  the 
general  equations  for  the  heat  absorbed  by  an  imperfect 
fluid,  and  for  convenience  are  brought  forward.  They  are 


dH=£pdr  -  r  (-^)  dp.  j 

In  the  first  of  these  equations  the  last  term  is  the  entire 
work,  both  external  and  internal,  due  to  an  expansion  dv, 
so  that  if  p'  be  such  a  pressure  that  when  multiplied  by  d  v 
would  equal  the  entire  work  done,  we  have 


(66) 


which  we  call  a  virtual  pressure.     In  Fig.  30,  if  vl  A  =  p 
be   the   external   pressure   at   the   vol- 
ume  v  and   temperature    r,  then   will 
some  ordinate,  as  vl  a,  represent  p ',  and 
hence 


^d  r> 

will  be  the  real  virtual  pressure,  being 

such  an  ideal  pressure  as  would  when 

multiplied  by  d  t  give  the  internal  work  due  to  expansion 

only.     If  the  path  A  B  be  arbitrary,  we  have,  generally, 


IMPERFECT  FLUIDS. 


[69.] 


v,  A  B  v,  =    lp  d  v. 
Vi  a  I  v,  =    fr  (d 


(r). 


»  +  V  W,         (67) 

the  indicated  integral  being  the  latent  heat  due  to  expan- 
sion, and  q>  (r)  a  function  of  the  temperature,  being  the 
latent  heat  due  to  an  increase  of  temperature. 

If  Kv  in  the  first  of  equations  ( A]  is  variable,  then  will  a 
part  of  the  heat  absorbed  do  internal  work  due  to  a  change 
of  temperature.  It  appears,  then,  that  the  internal  work 
may  be  considered  in  two  distinct  parts  :  one  due  entirely 
to  change  of  volume,  the  other  entirely  to  change  of 
temperature. 

69.  Let  the  temperature  be  constant  during 
expansion.  This  is  a  case  of  isothermal  expansion,  and  we 
have  d  r  =  o  and  T  =  r,,  and  the  first  of  equations  (A)  be- 


FIG.    31. 


(68) 


in  which  (—f. }    is  to  be  found  from  the 
\drJv 

equation  to  the  gas.  In  Fig.  31  let  a  b 
be  the  path  of  the  fluid,  which  will  be 
an  isothermal  of  the  substance,  then  will 
i>,  be  the  external  work  done  dur- 


ing  expansion,  and  let  aefb  represent  the  internal  work, 
then  will  vl  eft\  represent  the  total  work  done,  and  will  be 
the  latent  heat  of  expansion  ;  and  we  have 


ordinate  to  e  f=p'  =  T,  !-y*-J 

*•/*.- ••/&*)<* 


[69.]  LET  TEMPERATURE   BE   CONSTANT.  87 


vtal>vt=    I  pdv, 


EXERCISES. 

1.  If  equation  (7)  be  the  equation  of  an  imperfect  gas, 
find  the  total  work  done  during  an   isothermal  expansion 
from  #,  to  2  vt.     (Use  equation  (68).) 

2.  If  a  &,  Fig.  31,  be  the  isothermal  of  the  gases  in  equa- 
tion (7),  and  c  d  what  it  would  be  when  «,  ft  and  y  are  each 
zero,  give  the  equations  of  a  &,  c  d,  eft  and  the  values  of 
vl  a,  a  c  and  c  e. 

3.  Given  equation  (7)  to  find  the  external  work  -y,  a  1>  v^ 
Fig.  31,  and  the  internal,  a  efl>.    (Equations  (69)  and  (70)). 

4.  If  the  equation  of  the  gas  be  p  v  =  12  r  —  —  ^,  show 

that  a  c  =  c  e,  Fig.  31,  a  1)  being  the  isothermal  of  the  gas, 
and  c  d  what  the  isothermal  becomes  when  I  is  zero. 

If  one  pound  of  carbonic  acid  gas  at  300°  F.  expand 
isothermally  from  10  cubic  feet  to  20  cubic  feet,  find  the 
total  work  done,  also  the  external  and  internal  work.  (Use 
equations  (6),  (69)  and  (70).) 

5.  What  will  be  the  total  internal  work  of    expanding 
two  pounds  of  carbonic  acid  gas  indefinitely  at  the  constant 

-temperature  of  200°  F.,  the  initial  volume  being  8  cubic 
feet  ?  (The  limits  of  integration  in  equation  (70)  will  be  oo 
and  8.) 

Ans.  364  foot-pounds. 

6.  The  initial  volume  of  a  pound  of  carbonic  acid  gas 
being  8  cubic  feet,  how  much  must  it  be  compressed  at  the 
constant  absolute  temperature  of  600°  F.,  so  that  the  inter- 
nal work  shall  equal  the  external  work  ? 

7.  If  equation  (4)  be  the  equation  of  the  gas,  in  which 


88  IMPERFECT   FLUIDS.  [70,  71.] 

1  2  3 

«  =  — j  #,  =  — jj  #„  =  —3?  find  the  equation  of  an  isother- 
-y  v  v 

mal,  the  external  work  done  during  an  isothermal  expan- 
sion, and  the  total  work  done. 
Ans.  for  the  total  work, 

Sr,  logv-*-  f-J — ,  +  -i?-i  --^-i,&c. 

v,       Lr,  <      rl  v?      r?  v*       TI  v,  ' 

7O.  Change  of  state  in  regard  to  aggregation.  Let 
the  temperature  and  pressure  be  constant,  required  the  heat 
absorbed. 

For  this  case  dr  =  o,  and  r  fy£\  will  be  independent  of 
v,  hence 


These  conditions  are  realized  during  three  physical 
changes — fusion,  vaporization  and  sublimation. 

71.  Latent  heat  of  fusion,  or  of  liquefaction. 
Substances  may  be  melted — changing  from  a  solid  to  a  liquid 
state — under  the  constant  pressure  of  the  atmosphere,  or 
other  pressure,  and  at  a  fixed  temperature  for  that  pressure ; 
and  during  this  change  of  state  heat  is  absorbed  which  does 
not  affect  the  thermometer,  and  hence,  according  to  the 
definition,  is  called  latent.  Its  value  can  be  found  only  by 
direct  experiment.  Having  this  value  of  //  for  any  sub 
stance,  which,  for  distinction,  call  Iff  (noticing  that/"  is  tho 
initial  letter  of  fusion),  we  may  find 

d±  =  r(v,-v±  (72) 

dp  Uf 

for  which  the  rate  of  change  of  temperature  per  unit  of 
pressure  may  be  calculated.  If  the  volume  vl  of  the  sub- 
stance in  the  initial,  or  solid  state,  exceeds  that  in  the  ter- 
minal, or  liquid  state,  vu  then  will  — —  be  negative,  and  the 

d  p 


[71.] 


LATENT   HEAT   OF   FUSION. 


temperature  of  fusion  will  be  lowered  by  an  increase  of 
pressure,  a  principle  first  pointed  out  by  Professor  James 
Thomson  (Edinburgh  Trans.^Vol.  XVL).  Water,  antimony, 
cast  iron  and  some  other  substances,  are  more  bulky  in  the 
solid  than  in  the  liquid  state  ;  and  the  melting  point  of  all 
such  substances  is  lowered  by  pressure. 

The  latent  heat  of  fusion  of  ice  is  144  B.  T.  JL,  as  deter- 
mined by  experiment  or  144  X  778  =  112032  foot-pounds ; 
and  this  is  the  work  which  must  be  expended  upon  one  pound 
of  ice  at  32°  F.  in  reducing  it  to  liquid  water  at  the  same 
temperature,  which  work  is  necessary  to  completely  break 
down  the  crystalline  structure  of  the  ice.  Conversely,  it  is 
the  equivalent  of  the  heat-energy  which  must  be  emitted 
from  a  pound  of  water  and  absorbed  by  surrounding  objects 
in  changing  water  from  the  liquid  to  the  solid  state  at  32° 
F.  Solids,  under  a  definite  pressure,  have  a  corresponding 
definite  melting  point,  or  point  of  fusion. 

The  following  are  some  examples  of  the  latent  heat  of 
fusion  : 


Melting  point 

Latent  heat  of  fusion. 

Deg.  Fahr. 

B.  T.  U.                         Foot-pounds  Hf. 

Ice      

32 

144*                      112032 

Zinc 

793 

50  6                       39367 

Sulphur  .... 
Lead 

224 

635 

16.9                       13148 
9  7                        7547 

Mercury  .... 
Tin  

-40 
455 

5.1                        3968 
{500  as  given  by  Rankine. 
26.6  "      "       "   Box  on  Heat, 

Spermaceti  .  . 
Cast  iron  .... 

124 

2000 

p.  13t 
j      148  (Eankine). 
1        46.4  (Box). 
233  as  given  by  Clements. 

*  Phil.  Mag.,  1871,  XLL,  182.    Peclet  found  135,  Person,  144.04,  and 
144  appears  to  be  the  most  reliable. 

f  We  have  not  determined  which  (if  either)  is  correct. 


90  IMPERFECT   FLUIDS.  [71.] 

If  the  specific  heat  of  water  were  constant,  144  pounds  of 
water  at  any  temperature  above  33°  F.  would  have  its  tem- 
perature reduced  one  degree  in  just  melting  one  pound  of 
ice  at  32°.  The  mixture  after  melting  would  reduce  the 
temperature  a  little  more. 

The  expansive  force  resulting  from  congealing  water  was 
well  illustrated  by  Major  Williams  in  1786,  at  Quebec,  Can- 
ada, by  filling  an  iron  shell  with  water  and  driving  an  iron 
plug  weighing  over  2£  pounds  into  the  fuse  hole,  and  sub- 
jecting it  to  an  out-door  temperature  of  —  18£°  F. ;  when, 
upon  freezing,  the  plug  was  fired  out  and  projected  over  400 
feet  (Trans.  Koy.  Soc.  Edinburgh,  II.,  23). 

EXERCISES 

1.  If  for  water  we  have 

r  =  492.66°  F. 

v,  =  0.01602  cu.  ft.  per  pound  ; 
and  for  ice 

r  =  492.66°  F. 

-y,  —  0.0174  cu.  ft.  per  pound, 
and  Ht  =  112032  ; 

how  much  will  the  melting  point  of  ice  be  lowered  by  a 
pressure  of  one  atmosphere,  2116.2  pounds  per  square  foot? 
(Use  equation  (72).) 

Ans.  0.0128°  F. 
0.0071°  C. 

2.  Kequired  the  pressure  per  square  foot  necessary  to 
lower  the  melting  point  of  ice  1°  F. 

We  have 

_&J>__         Hf  112032 

d  r  ~  T  (w,  -  t>.)  ~  492.66  X  0.00138  ~  164784  Ibs" 
In  this  exercise  d  T  is  considered  unity  and  dp  =  164784  ,•  or  the  sec- 
ond member  may  be  considered  constant  and  the  left  member  integrated 
between  limits,  giving  p*  ~P'=  164784.    The  notation  of  the  preceding 
exercise  may  be  treated  in  a  similar  manner. 


[72.]  EXPERIMENTAL   VERIFICATION.  91 

3.  If  1140  pounds  pressure  per  square  inch  will  lower  the 
melting  point  of  ice  from  32°  F.  to  31°  F.,  diminishing  the 
volume  of  one  pound  0.00138  of  a  cubic  foot ;  required  the 
latent  heat  of  fusion  of  one  pound  of  the  ice. 

Ans.  143.2. 

4.  Required  the  external  and  internal  work  in  melting  ice 
at  32°  F.  at  atmospheric  pressure. 

The  external  work  will  be  that  done  in  lowering  the  atmosphere  through 
a  distance  equal  to  the  decrease  of  volume  in  changing  the  state  of  ag- 
gregation, or  2116.3  X  0.00138  =  3  foot-pounds,  nearly.  The  total  work 
will  be  by  using  the  result  in  exercise  2, 

T  ^dv  =  492.66  X  164784  X  0.00138  =  112032  ft.-lbs.,  nearly, 
dr 

and  3  pounds  more  due  to  atmospheric  pressure. 

From  this  it  appears  that  the  work  is  nearly  all  internal,  and  is  more 
than  36500  times  the  external  work. 

5.  The  pressure  required  to  reduce  the  melting  point  of 
ice  1°  F.  being  16-4784  pounds  per  square  foot  when  the  ini- 
tial temperature  is  r  =  492.66°  F. ;  find  the  diminution  of 
volume  of  one  pound  in  changing  from  congealed   to  liquid 
water. 

6.  Required  the  pressure  necessary  to  reduce  the  melting 
point  of  ice  to  —  18°  C.,  assuming  that  the  above  formula 
is  valid  so  far  below  0°  C. 

7.  What  is  the  highest  temperature  at  which  ice  can  exist 
indefinitely  in  a  vacuum  ? 

72.  Experimental  verification.  Sir  William 
Thomson,  by  a  delicate  and  beautiful  experiment,  proved 
that  the  melting  point  of  ice  was  lowered  by  pressure  (Phil. 
Mag.  (1850),  III.,  XXXVII.,  123).  A  delicate  thermometer, 
constructed  for  the  purpose,  was  enclosed  in  a  vessel  with 
water  and  lumps  of  clear  ice  and  an  air  gauge  for  measuring 
the  pressure.  At  atmospheric  pressure  the  ice  would-  not 
melt  if  below  32°  F.,  but  it  was  found  that  when  the  con- 


92  IMPERFECT   FLUIDS.  [73.} 

tents  of  the  vessel  was  subjected  to  pressure  the  thermome- 
ter fell  as  the  water  assumed  the  temperature  of  the  melting 
point  of  ice  corresponding  to  the  increased  pressure  ;  and 
the  observed  results  corresponded  well  with  those  calculated. 
Professor  Mousson  made  the  following  experiment  :     A 
prism  of  steel,  Fig.  32,  was  used,  having  a 
cylindrical  bore  0.71  cent.  (0.28  inch),  closed 
at  the  lower  end  by  a  copper  cone  forced  in 
by  a  strong  screw,  and  the  upper  end  by  a 
long  slightly  conical  copper  plug  a  pressed 
down  by  a  steel  piston  by  means  of  a  strong 
screw,  and  when   in  an  inverted   position 
a  small  brass  rod  b  was  dropped  in  and  the 
FIG.  32.  bore  filled  with  water.  After  being  exposed 

to  cold  at  —  9.5°  C.  the  protruding  ice  was 
removed,  the  copper  cone  inserted  and  screwed  up,  and  the 
whole  reversed  and  put  into  a  freezing  mixture  at  —  18° 
C.,  after  which  the  upper  plug  was  forced  in  at  a  pressure 
roughly  estimated  at  not  less  than  13250  atmospheres.  When 
the  lower  plug  was  removed  the  brass  rod  dropped  out  first, 
showing  that  the  ice  had  been  melted,  permitting  the  rod  to 
fall  to  the  lower  end.  The  pressure  was  more  than  five 
times  that  required  by  theory  to  melt  the  ice,  but  the  tem- 
perature at  which  it  melted  is  unknown. 

73.  It  is  natural  to  infer  \\iQOppositeprinciple — that  the 
melting  point  of  those  substances  which  expand  when  fused 
will  be  raised  by  compression,  and  this  principle  has  been 
verified  by  Mr.  Hopkins  (Rep.  B.  A.  (1854),  II.,  56),  as  well 
as  by  others.  In  Mr.  Ilopkins's  experiments  the  instant  of 
fusion  was  determined  by  means  of  a  small  iron  ball  sup- 
ported by  the  substance  when  solid,  but  which  fell  when  the 
substance  liquefied  ;  and  when  supported  it  deflected  a  needle 
which  was  suspended  just  outside  the  vessel,  but  the  deflection 
ceased  when  the  ball  fell.  The  temperature  was  determin- 
ed by  that  of  the  oil  in  a  bath  in  which  the  whole  was  im- 


[73.] 


EXPEKIMENTAL   VERIFICATION. 


93 


mersed ;  and  the  effective  pressure  was  taken  as  the  half 
sum  of  the  pressures  which  forced  the  piston  inward  and 
that  required  to  just  permit  it  to  return  to  its  initial  position, 
tims  elminating  the  effect  of  friction. 
The  following  results  were  found  : — 


Melting  points  in  degrees  centigrade. 

Pressure 

Atmospheres. 

Spermaceti. 

Wax. 

Sterrane. 

Sulphur. 

1 

51.1 

64.7 

67.2 

107.2 

538.4 

60.0 

74.7 

68.3 

135.3 

820.5 

80.3 

80.3 

73.9 

140.6 

The  melting  point  of  wax  will  be  given  nearly  by  the  em- 
pirical formula 

te  =  64.68  +  0.0188  p. 

M.  Person  gives  the  following  empirical  formula  as  the 
results  of  his  experiments  on  the  latent  heat  of  fusion  of 
non-metallic  substances  (Ann.  de  Chem.  et  de  P/iys.,  Nov., 
1849). 

I  =  (c1  -  c)  (T  +  256°)  (73) 

in  which 

c  =  the  specific  heat  of  the  substance  in  the  solid  state, 
o'  =    «         "         "         «  "     .        "     liquid  state, 

T  =  its  temperature  of  fusion  in  degrees  Fahr.,  and 
I  =  the  latent  heat  of  fusion  of  one  pound  in  B.  T.  IT. 
at  the  pressure  of  the  atmosphere. 


EXERCISES. 


1.  If  the  specific  heat  of  water  be  unity,  of  ice  0.504,  and 
the  temperature  of  fusion,  T7,  be  32°  F.,  required  the  latent 
heat  of  fusion. 

2.  If  ice  be  subjected  to  such  a  pressure  that  it  will  melt 


94  IMPERFECT   FLUIDS.  [74.] 

at  0°  F.,  find  its  latent  heat  of  fusion,  the  law  being  assumed 
to  hold  good  to  that  point ;  and  the  amount  less  than  at 
32°  F. 

3.  What  must  be  the  temperature  of  ice  that  there  will  be 
no  latent  heat  of  fusion,  and  to  what  pressure  must  it  be 
subjected  ? 

Ans.  T  =  -  256°  F. 

Pressure  =  256  +  3-  +  1  =  22500  atmospheres. 
0.0128 

If  this  could  be  realized  there  would  be  no  mechanical  distinction  be- 
low this  temperature  between  solid  and  liquid  water. 

4.  Assuming  that  experimental  results  may  be  used  to  the 
extent  implied  in  the  questions  here  given,  find  the  increase 
of  volume  due  to  the  fusion  of  spermaceti  at  124°  F. 

For  the  ratio  of  d  p  to  d  T  considered  as  finite,  see  the  preceding  table  ; 
then  we  have 

Ik    _      148  X  778 

Vt       Vl  "     dp-  537.4X2116' 

Td~r  *    140-124 

5.  The  latent  heat  of  fusion  of  solid  water  at  32°  F.  being 
144  B.  T.  U.  and  its  specific  heat  in  the  liquid  state  being 
unity,  find  its  specific  heat  in  the  solid  state.    (Equation  (73).) 

6.  The  specific  heat  of  ice  being  0.504  at  32°  F.,  and  the  la- 
tent heat  of  fusion  144  B.  T.  U.,  find  the  specific  heat  of  water. 

7.  The  specific  heat  of  solid  spermaceti  being  0.32  and 
considering  its  latent  heat  of  fusion  as  46.4  B.  T.  U.  at 
120°  F.,  find  its  specific  heat  in  the  liquid  state.  (Equation 
(73).) 

74.  Latent  heat  of  evaporation.  It  is  found  by 
experiment  that  there  is  a  definite  boiling  point  for  liquids 
corresponding  to  the  pressure  to  which  they  are  subjected, 
and  from  this  condition  they  will  pass  into  a  vapor  at  that 
temperature  and  pressure.  Hence,  equation  (71)  is  direct- 
ly applicable  to  this  case,  and  indicating  this  particular  latent 


[74.]  LATENT  HEAT   OF   EVAPORATION.  95 

lieat  by  I16  (e  being  the  initial  letter  of  evaporation),  we  have 


and  J  Ae  =  Hv 

By  determining  the  factors  of  equation  (74)  by  experi- 
ment, the  value  of  I16  may  be  computed.  M.  Regnault  de- 
termined the  value  of  Ae  directly  for  water  at  a"  series  of 
boiling  points  from  its  freezing  point  to  about  375°  F.,  which 
may  be  represented  with  great  precision  by  the  empirical 
formula,  in  English  units, 

JJe  =  [1091.7  -  0.695  (T-  32)  -  0.000000103  (T  -  39.1)3  ]  X  778,  (75) 
or  in  French  units, 
7/e  =  [606.5  —  0.695  T  -  0.00000033  (T  —  4)3  ]  426.8.  (76) 

(Me'moire  Academie  des  Sciences,  1847.  Trans.  Roy. 
Soc.,  Edinburgh,  Yol.  XX.) 

In  practice  it  will  be  sufficiently  exact  to  use  the  follow- 
ing:— 

£e  =  966  —  0.7  (T  —  212) 
=  1092  -  0.7  (T  -  32) 
=  1114.4  -  0.7  T 

=  1436.8  -  0.7  T,  B.  T.  U.,  (77) 

or  its  equivalent, 

//e  =  751548  -  544.6  (T  -  212) 
=  867003  -  544.6    T 
=  1117880  —  544.6  T,  foot-pounds,      (78; 
=  a  —  1)  r. 

The  latent  heat  of  evaporation  of  some  other  substances 
is  given  in  the  Addenda. 

The  exact  value  of  the  latent  heat  of  evaporation  of  one 
pound  of  water  at  the  pressure  of  one  atmosphere,  as  found 
by  Eegnault,  is  966.1  B.  T.  U.  =  751624  foot-pounds. 
This  is  the  work  necessary  to  simply  change  water  from  its 
liquid  state  when  at  212°  F.  under  the  pressure  of  one  at- 


96 


IMPERFECT   FLUIDS. 


L'3-i 


mosphere  to  the  condition  of  vapor  at  the  same  temperature 
and  pressure.  Since  water  in  the  form  of  steam  occupies 
more  space  than  as  a  liquid,  the  molecules  must  be  farther 
apart  in  the  former  than  in  the  latter  state,  and  hence,  with- 
out considering  their  exact  condition  in  the  two  cases,  it  ap- 
pears that  it  requires  751624  foot-pounds  of  energy  to  sim- 
ply separate  the  molecules  of  a  pound  of  water  sufficiently  to 
produce  steam.  This  amount  of  heat  is  absorbed  and  dis- 
appears without  affecting  the  temperature  ;  and  the  same 
amount  reappears,  or  passes  to  other  bodies,  when  it  returns 
to  a  liquid  at  that  pressure. 


Latent  heat  of  evaporation  of  one  pound  of  certain  substances  at  the  pressure  of 
one  atmosphere. 


Substance. 

Boiling  point. 
Deg.  F. 

Latent  heat. 

B.  T.  U. 

Foot-pounds. 

Water     

212 
172.2 
95.0 
114.8 
316 
141 

966.1 

364.3 
162.8 
156.0 
184.0 
236.0 

751624 
283425 
125658 
121368 
143152 
l.«3608 

Alcohol 

Ether 

Bisulphide  of  carbon  
Oil  of  turpentine 

Naphtha 

75.  Vapor  is  any  substance  in  a  gaseous  condition  at 
the  maximum  density  for  that  tempfffittire  or  pressure.  If  a 
vapor  be  in  contact  with  the  liquid,  as  steam  in  the  same 
vessel  as  water,  there  is  a  certain  pressure  which  is  at  once 
the  least  pressure  under  which  the  substance  can  exist  in  the 
liquid  state  and  the  greatest  pressure  at  which  it  can  exist  in 
the  gaseous  state  at  that  temperature.  Under  such  condi- 
tions the  vapor  is  called  saturated  vapor  (or  saturated  steam\ 
and  the  pressure,  the  pressure  of  saturation.  If  at  the  tem- 
perature of  saturation  the  pressure  be  slightly  increased  some 


[76.]        PRESSURE  AND  TEMPERATURE  OF  A  VAPOR.         97 

of  the  vapor  will  be  condensed  until  equilibrium  be  restored, 
and,  conversely,  if  the  pressure  be  diminished  more  vapor 
will  be  generated.  To  illustrate,  if  a  cylinder  containing  a 
piston  were  placed  directly  over  a  steam  boiler  having  one 
end  open  to  the  boiler,  and  the  piston  be  forced  inward,  a 
portion  of  the  steam  would  be  condensed,  and  if  the  piston 
be  drawn  outward  more  steam  would  be  generated  —  the  tem- 
perature and  pressure  remaining  constant  while  the  volume 
varied.  This  is  known  as  Dalton's  law  (Stewart  on  Heat, 
p.  143). 

76.  The  relation  between  pressure  and  tem- 
perature of  a  vapor  can  be  determined  only  by  experi- 
ment, as  has  been  done  by  Regnault  (Memoirs  de  V  Acade- 
mie  des  Sciences,  1847  ;  Comptes  Rendus,  1854).  His 
results  are  represented  quite  accurately  by  the  following 
empirical  formula,  given  by  Rankine,  and  first  published  in 
the  Edinburgh  New  Philosophical  Journal  for  July,  1849 
(Phil.  Mag.,  Dec,,  1854). 

com.  log  p  =  A.  —  --  —  (80) 

where  A,  B,  C,  are  constants  to  be  determined  by  experi- 
ment. That  author  remarks  that  this  formula  is  sufficiently 
accurate  for  temperatures  from  —  22°  F.  to  446°  F.  From 
(80)  we  find 

'  (81) 

B 


=  p         .  +  X  2.3026,  (82) 


X  2.3026.  (83) 

The   following  are  the  values  of  the  constants  for  the 


98  IMPERFECT  FLUIDS.  [77.] 

vapor  of  water  (saturated  steam)  for  degrees  Fahrenheit  and 
pressures  \npounds  per  square  foot  : 

A  =  8.28203,  log  B  =  3.441474,  log  C  =  5.583973. 

These  values  differ  slightly  from  those  given  by  Rankine, 
because  he  used  r0  =  493.2°  F.,  while  here  r0  =  492.66°  F. 
Kegnault's  experiments  also  furnish  the  data  for  determin- 
ing the  constants  for  Alcohol,  Ether,  Bisulphide  of  Carbon 
and  Mercury. 

7  7 .  Volume  of  vapor.  From  equations  (74)  and  (83) 
we  have 


V.  —  V,    = 


rdp  (84) 

d  r 


which  for  saturated  steam  becomes,  equation  (78), 
1117880  -  544.6  r 


MT  +  - 

from  which  ra,  the  volume  of  a  pound  of  the  steam,  may  be 
determined.  It  will  be  found  hereafter  that  the  volume  of 
a  pound  of  saturated  steam  at  212°,  the  pressure  of  one  at- 
mosphere being  29.9218  inches  of  mercury,  is  26.50  cubic 
feet,  nearly,  and  the  volume  of  one  pound  of  water  at  its 
maximum  density,  39.1°  F.,  is  0.01602  cubic  feet ;  hence,  a 
pound  of  saturated  steam  under  the  pressure  of  one  atmos- 
phere occupies  1654  times  the  volume  of  one  pound  of 
water  at  its  maximum  density.  This  increase  is  illustrated 
by  Fig.  33,  in  which  the  small  square  at  the  upper  left-hand 
corner  represents  the  volume  of  one  pound  of  water, 
i\  —  0.01602,  and  the  entire  large  square  the  volume  occu- 


178.] 


WEIGHT   OF   VAPOR. 


99 


pied  by  the  steam  produced  from  it  at  212°  F.,  v%  =  26.58,  and 
the  large  square 
minus  the  small 
one  represents  the 
increase  of  vol- 
ume in  changing 
the  water  to  va- 
por, Vy  V^  If 

the  boiling  point 
increases  in  tem- 
perature, the  vol- 
ume of  steam  de- 
creases ;  still  with- 
in the  range  of 
ordinary  practice 
the  volume  of 
water  is  so  small  FIG-  33- 

compared  with  that  of  the  steam  generated  from  it,  the  former 
may  be  neglected,  and  we  have  with  sufficient  accuracy 


(86) 


78.  Weight  of  vapor  per  cubic  foot,  sometimes 
called  the  density  of  the  vapor.  Since  a  pound  of  the  vapor 
occupies  <y2  cubic  feet,  the  weight  of  a  cubic  foot  of  the 
vapor  will  be 

1 
w  =  — 

^2 

If  L  be  the  latent  heat  of  evaporation  per  cubic  foot,  then 


d  r 


(87) 


which  is  the  reciprocal  of  equation  (84). 


100 


IMPERFECT   FLUIDS. 


[78.] 


A  series  of  values  of  pressures,  J9,  and 
volumes,  v,  may  be  calculated  by  means  of 
equations  (80)  and  (85),  according  to  the 
method  shown  in  the  following  exercises  5 
and  6  ;  the  results  of  which  may  form  a  table 
like  the  one  at  the  end  of  this  volume  for 
"  Saturated  Steam."  From  these  results  the 
heavy  line  in  Fig.  33«  has  been  constructed, 
in  which  the  abscissas  are  volumes  per  pound, 
and  the  ordi nates,  pressures  per  square  inch. 
Such  a  curve  is  called  the  curve  of  satura- 
tion. 

Assuming  its  equation  to  be 


JO  40  1 

•0      Gl 

>     7U     W 

•0   1M 

lip, 

md  p 
or,  p 
or,p 
or,p 

PIG. 

=  14. 

=  60, 
=  100, 
=  160, 
=  200, 

33a. 

7,  v, 

v 

V 
V 

=  26.59, 
=  7.100, 
=  4.403, 
=  2.830, 
=  2.294, 

then  n  = 

n  = 
n  = 
n  = 

1 
1 
1 
1 

.06470  ; 
.065837  ; 
.065954  ; 
.06551. 

Mean  n  =  1.06550  ; 

.-.pv1-'""  =  constant,  (88) 

or,  p  tfl  =  484,  (89) 

p  being  pounds  per  square  inch  and  v  cubic  feet  per  pound. 

EXERCISES. 

1.  Required  the  latent  heat  of  evaporation  of  water  at 
20°  F.,  60°  F.,  200°  F.,  400°  F.,  and  explain  why  it  is  less 


[78.]  WEIGHT   OF   VAPOK.  101 

at  the  higher  temperatures.  (Equation  (78),  or  tables  of 
saturated  steam.) 

2.  If  Kegnault's  law  can  be  trusted  so  far,  find  the  tem- 
perature at  which  the  latent  heat  of   evaporation  will  be 
zero  for  steam. 

Ans.    T  =  2052°  F.  or  T  =  1592°  F., 

or  about  the  temperature  of  the  melting  point  of  brass, 
above  which  there  would  be  no  difference  between  the 
liquid  and  vaporous  forms.  This  is  called  the  critical  tem- 
perature, and  has  been  determined  experimentally  for  some 
substances  (Phil.  Trans.  (1869),  CLIX.,  575). 

3.  If  the  latent  heat  of  evaporation  of  one  pound  of  water 
at  the  melting  point  of  ice  could  be  utilized  in   projecting 
that  pound  vertically  upward,  how  many  miles  high  would 
it  ascend  in  a   vacuum,  considering  gravity  constant,   and 
g  =  32.2  ? 

4.  Through  what  height  must  a  100-pound  ball  descend 
in  a  vacuum  so  that  its  energy  if  entirely  utilized  for  the 
purpose  would  just  evaporate  one  pound  of  water  at  and 
from  212°  F. 

5.  Find  the  value  of  JL  for  saturated  steam  at  212°  F. 

d  r 

If  we  resort  to  tables  of  saturated  steam,  we  find  that  Rankine's  Table 
VI.  (Steam  Engine)  is  not  suitable  for  this  purpose,  because  the  tempera- 
tures are  for  differences  of  9°  F.  Several  other  tables  give  the  pressures 
for  every  degree.  From  one  of  these  we  find  that  at 

211°  F.,  the  pressure  is  14.406  Ibs.  per  sq.  in. 
212°  ..       "        «         «  14696    "      "     "     " 
213°  "       "         "         "  14.991    "      "     "     " 
Hence,  from  211°  to  212°  we  have 

~  =  0.290  X  144  =  41.76,  approximately, 
from  212°  to  213°,  we  have 

=  0.295  X  144  =  42.48*  approximately, 


102  IMPERFECT   FLUIDS.  [78.] 

the  mean  of  which  gives 


which  is  nearer  correct,  but  cannot  be  exact.    To  find  the  correct  value 
directly,  use  equation  (82),  and  we  have 

dp*     2116.2  r     B  2(7    "I         3026 

d  T  ~  672.66  L  672. 66       (672.66)'2J 

-M1UX  M«X|^ 

=  42.06, 
which  is  the  value  required. 

6.  Find  the  volume  of  a  pound  of  saturated  steam  at 
212°  F.,  and  the  weight  of  a  cubic  foot  of  it. 
•  From  equation  (84)  we  have 

ffe 
*'  *%P 

(I  T 

Water  increases  in  volume  0.04775  per  unit  from  that  at  its 
maximum  density  to  212°  F.  ;  hence, 

v,  =  0.01602  X  1.04775  =  0.01678  =  0.017 

with  sufficient  accuracy. 
Then 

«.  =  0.017  +  pffvvYlp  =  26'585  CU" ft" 
6  <  2.66  X  42.06 

and 

w  =  —  =  0.03762  Ibs.  per  cu.  ft. 

(Rankine  gives  the  following  empirical  formula  for  the  vol- 
ume of  liquid  water  at  any  absolute  temperature. 


If  steam  followed  the  law  of  perfect  gases,  we  could  now 


[78.]  WEIGHT   OF   VAPOR.  103 

find  the  volume  of  a  pound  of  it  at  any  temperature.     For 
we  would  have 

pv      2116.2  x  26.5 

•      -    ~~ 


.-.*=  83.37     -  (89a) 

7.  Find  the  volume  of  one  pound  of  saturated  steam  at 
1200°  F. 

8.  Find  the  volume  of  saturated  steam  at  212°  F.  gener- 
ated from  one  cubic  foot  of  water. 

If  fla  =  26.6,  as  in  Exercise  6,  then  will  the  required  volume  be  1659 
cubic  feet.  Heretofore  writers  have  used  Rankine's  results,  giving  1644 
or  1645.  Observation  has  not  fixed  the  exact  value. 

9.  Find   the   external  and  internal  work  in  changing  one 
pound  of  water  into  steam  at  and  from  212°  F.,  and  their 
ratio. 

The  volume  of  one  pound  of  steam  will  be        vy  =  26.585, 
"      "      "        "      "    water  at  212°  F.,  vi  =        017, 

«2  -  «,  =  26.568. 
The  entire  work  will  be 

r  j£  (v,  -  v,)  =  672.66  X  42.06  X  26.568  =  751672. 
966.1  X  778  =  751626. 

These  results  would  be  identical  if  all  the  quantities  were  determined 
with  precision. 

The  external  work  consists  in  forcing  the  pressure  of  one  atmosphere 
through  26.568  feet,  or 

2116.2  X  26.568  =  56219  foot-pounds, 
which  deducted  from  the  former,  gives 

751672  —  56219  =  695453  foot-pounds 

for  the  internal  work  ;  hence,  the  internal  work  will  be 

695453 


56219 
times  the  external  for  the  conditions  given  in  the  problem. 

78a.  Isothermal  of  a  liquid,  its  vapor  and  its 
gas. 


104  IMPERFECT    FLUIDS.  [79-] 

To  illustrate,  conceive  that  one  pound  of  the  liquid  is 
subjected  to  an  immense  pressure  at  some  moderate  tem- 
perature, T,  —  say  2 1 2°  F.  Let  A ,  Fig.  d, 
represent  this  state,  the  ordinate  to  which 
will  represent  the  pressure,  and  the  ab- 
scissa the  volume.  Let  the  pressure  be 
gradually  removed  while  the  tempera- 
ture is  maintained  constant  — the  volume 
will  increase  slightly  and  the  path  of  the 
fluid  may  be  represented  by  A  B,  the 
abscissa  of  £  exceeding  that  of  A. 
Let  B  be  the  state  at  which  the  liquid  will  boil  at 
the  temperature  r ;  then  at  that  pressure  the  volume 
will  increase,  the  temperature  remaining  constant,  and 
B  C  will  represent  the  path.  At  C  let  the  pound  be 
entirely  vaporized  and  the  pressure-gradually  removed,  while 
the  temperature,  T,  is  maintained  constant ;  then  will  the 
path  be  the  isothermal  C  r.  The  entire  broken  line  A  B  C  r 
is  an  isothermal  of  the  fluid.  Let  the  operation  be  re- 
peated at  a  higher  temperature — the  boiling  point  will  be 
reached  at  a  state  above  and  a  little  to  the  right  of  B,  so  that 
a  curve  F  E  passed  through  such  points  may  be  called  the 
locus  of  boiling  points.  Continuing  the  operation  as  be- 
fore, and  the  state  at  which  the  pound  will  be  evaporated 
will  be  represented  by  a  point  above  and  to  the  left  of  C, 
and  the  curve  traced  through  all  such  points  will  be  the 
curve  of  saturation  E  C  G,  already  described.  These 
curves  will  meet  at  some  point  E,  and  the  temperature  of 
that  state  is  called  the  critical  temperature. 

79.  An  experimental  determination  of  the 
density  of  saturated  steam  was  first  made  by  Fairbairn  and 
Tate  in  1860  (Phil.  Trans.  (London),  CL.,  (1860),  185; 
CLIL,  (1862),  591).  The  densities  as  thus  found  differed 
from  those  previously  found  by  Rankine  from  T£F  to  ^  of  the 
experimental  values,  thereby  giving  larger  values  above  242° 


[80.]  MEASUKEMENT   OF   HEIGHTS.  105 

F.,  and  below  some  were  larger  and  others  smaller  than  the 
experimental  ones  (Miscellaneous  Scientific  Papers,  p.  423). 
In  these  experiments,  the  steam  was  in  a  statical  condition, 
while  in  Regnault's  experiments  the  steam  was  in  rapid  mo- 
tion from  the  boiler  to  the  condenser — differences  of  con- 
dition which  would  naturally  affect  the  results.  The  proper 
value  of  J  would  also  affect  the  result ;  but  the  results  ob- 
tained by  the  two  methods  agree  as  nearly  as  one  might  ex- 
pect under  the  circumstances.  Rankine's  Tables  are  in  his 
"  Prime  Movers,"  and  those  of  Fairbairn  and  Tate,  above 
the  pressure  of  one  atmosphere,  are  in  Richard's  Steam  In- 
dicator, Weisbach  on  the  Steam  Engine,  and  other  works. 

The  apparatus  employed  by  Fairbairn  and  Tate  for  deter- 
mining the  temperature  of  saturation  consisted  of  two  glass 
globes  connected  by  a  bent  tube  below  them.  The  tube  was 
filled  with  mercury,  above  which  in  the  globes  was  the 
liquid,  one  containing  more  than  the  other,  then  the  globes 
and  tube  were  placed  inside  of  a  small  boiler  containing  the 
same  liquid,  and  the  whole  heated.  So  long  as  the  steam  is 
saturated  the  mercury  in  the  tube  will  remain  stationary,  but 
the  instant  that  the  smaller  volume  of  water  is  all  changed 
to  vapor  (some  water  still  remaining  in  the  other),  the  mer- 
cury will  rise  in  that  end  of  the  tube  nearest  the  globe  in 
which  all  the  water  has  been  evaporated,  as 
after  that,  that  steam  becomes  superheated, 
and  the  rate  of  increase  of  pressure  is 
for  saturated  than  for  superheated  vapor.  At 
the  instant  of  change  the  volume  of  the 
steam  will  be  the  volume  of  the  space  above 
the  mercury,  and  the  temperature  and  pres- 
sure of  the  steam  in  the  globes  will  be  the  same  as  that  in 
the  boiler,  and  hence  may  be  readily  measured. 

8O.  Measurement   of  heights.      The    principles 

*  Theori*  Mecanique  de  la  Chaleur,  %™  ed.  (1875),  II. 


106  IMPERFECT  FLUIDS.  [81,  82.] 

here  discussed  furnish  the  means  of  determining  altitudes. 
Thus,  if  at  any  point  on  a  hill  or  mountain  the  temperature 
of  boiling  water  be  observed,  the  pressure  of  the  atmosphere 
at  that  place  and  time  may  be  computed  by  means  of  equa- 
tion (80),  then  will  the  height  above  the  level  of  the  sea  be 

h  =  60346  log  ^  feet,  (90) 

in  which  pQ  is  the  pressure  of  the  atmosphere  at  sea-level, 
2116.2  pounds  per  square  foot. 

(See  the  author's  Elementary  Mechanics,  p.  328.) 


8  1 .  Sublimation  consists  of  a  change  from  the  solid 
to  the  vaporous  state  without  passing  through  the  liquid 
state.  Experimental  data  in  regard  to  this  change  are  want- 
ing, so  that  we  are  unable  to  make  use  of  any  analysis  rep- 
resenting this  change. 

82.  Evaporation  without  ebullition.  Experi- 
ment indicates  that  evaporation  takes  place  at  all  temperatures 
for  a  great  variety  of  substances,  if  not  for  all.  Snow  and  ice 
evaporate  at  temperatures  below  freezing,  and  many  solids 
emit  sufficient  vapor  to  be  detected  by  the  sense  of  smell. 
The  evaporation  of  water  in  the  atmosphere  is  the  most  im- 
portant part  of  this  subject.  Water  is  elevated  in  the  form 
of  mist  in  the  atmosphere,  forming  clouds,  by  which  means 
water  is  redistributed  over  the  earth.  Evaporation  takes 
place  at  the  surface  of  bodies  of  water  and  is  unaffected  by 
the  conditions  below,  except  so  far  as  they  modify  the  sur- 
face. It  is  generally  greatest  at  higher  temperatures,  al- 
though other  conditions  modify  this  law.  It  is  modified  by 
the  hygrometrical  conditions  of  the  atmosphere,  and  is  greater 
during  a  wind  than  during  a  calm.  It  is  greater  during 
summer  than  winter,  and  generally  greater  during  June  and 


[83.]        PKESSURE   AND   VOLUME   BOTH   CONSTANT.         107 

July  than  for  any  other  months  of  the  year,  but  it  is  some- 
times, though  rarely,  greater  during  August  than  for  any 
other  month.  It  is  greatly  affected  by  locality,  there  being 
sometimes  a  difference  of  several  inches  within  a  few  miles. 
It  has  been  observed  that  one  half  an  inch  in  depth  has  been 
evaporated  from  a  large  pond  in  twenty-four  hours  in  lati- 
tude about  42°  north,  but  this  is  an  extreme  case.  The 
amount  is  also  very  different  for  different  years,  the  maxi- 
mum exceeding  the  minimum  by  more  than  fifty  per  cent. 
Twenty  to  thirty -five  inches  of  evaporation  per  year  is  very 
common,  in  which  cases  the  amount 

for  June  would  range  from  about  2.8  inches  to  5.4  inches, 
"  July         "  "         "  "     3.0      "       "  5.8        " 

"  August    "  "         '<  "     2.3      «       "  5.3        " 

It  is  recorded  that  in  July,  1875,  near  Boston,  the  evapo- 
ration was  7.21  inches.  (Much  valuable  information  on  this 
subject  was  collected  and  published  for  the  use  of  the  court 
in  The  Case  in  the  Supreme  Court  of  the  State  of  New 
York — General  Term — Fifth  Department — for  an  Applica- 
tion by  the  City  of  Rochester  to  acquire  certain  rights  to 
draw  water  from  Hemlock  Lake,  Yol.  II.,  about  1884. 
Also  certain  Transactions  of  the  American  Society  of  Civil 
Engineers,  New  York,  particularly  a  paper  by  Mr.  D.  Fitz- 
gerald, Transactions,  Yol.  XV.,  581,  (1886).) 

83.  Assume  the  pressure  and  volume  both 
constant  during  the  absorption  of  heat.    These  conditions 
make  dp  =  o,  and  d  v  =  0,  in  equations  (A) ;  hence, 
d  H  =  Kv  d  r  =  Kv  d  r ; 
.-.  K,  =  Kv. 

Strictly  speaking,  these  conditions  are  realized  for  only  a 
few  exceptional  cases,  as  water  at  its  maximum  density. 
Generally,  the  volume  changes  during  the  absorption  oi 
heat  under  constant  pressure,  but  for  solids  and  liquids,  the 
change  of  volume  under  constant  pressure,  and  of  pressure 
at  constant  volume  is  so  small  they  may.be  considered  as 


108 


IMPERFECT  FLUIDS. 


[84.] 


constant,  and  more  especially  so  when  it  is  considered  that 
the  work  due  to  these  changes  for  these  conditions  is  usually 
very  small  compared  with  the  energy  expended  in  making 
the  substance  hot.  If  C  be  the  mean  specific  heat  of  a  solid 
or  a  liquid  for  a  large  range  of  temperatures,  we  have  prac- 
tically, 

JTV  =  KV  =  a 

Tables  of  specific  heats  of  solids  and  liquids  give  values 
for  C  only. 

Specific  heat  of  a  few  solids  and  liquids. 


Snbstance. 

Specific  heat 
C. 

Authority. 

Wrought  iron  

0.11379 

Ttegnault. 

Cast  iron 

0  12983 

« 

Copper 

0  09511 

« 

Ice  

0.504 

Person. 

Spermaceti 

0.320 

Irvine. 

Alcohol 

0622 

Dalton 

Sulphur  

0.20259 

Regnault. 

84.  Mechanical  mixtures  may  produce  a  change  of 
the  state  of  aggregation,  as  when  warm  water  melts  ice.  The 
general  equations  of  thermodynamics  are  discontinuous  for 
such  cases  ;  but  having  determined  expressions  for  the  heat 
absorbed  by  fusion  and  evaporation,  we  may  write  an  ex- 
pression for  the  heat  absorbed  in  passing  from  the  solid  to 
the  gaseous  state.  Thus,  let  <?,  be  the  specific  heat  of  the 
substance  in  the  solid  state,  c3  of  the  liquid,  cs  of  'the  vapor, 
w  the  weight  of  the  substance,  T0  the  temperature  of  the 
melting  point,  T7,  of  the  boiling  point,  T'  the  initial  and  T" 
the  final  temperatures  ;  then  will  the  heat  absorbed  in  passing 
from  a  temperature  T'  below  freezing  to  T"  above  boiling  be 


which  for  one  pound  of  water  at  the  pressure  of  one  atmos- 
phere. becomes 


[84.]  MECHANICAL    MIXTURES.  109 

h  =  0.504(32  -  T.)  4-  144 4- 180  4-  966  4-  0.48 (T  —  212°) 
=  1204.4  -  0.504  T  4-  0.48  T  (92) 

=  1200  +  0.5  (T  •  -  T'\  nearly. 

EXERCISES. 

1.  Required  the  temperature  after  mixing  3  pounds  of 
water  at  90°  F.,  10  pounds  of  alcohol  at  30°  F.,  and  20 
pounds  of  mercury  at  60°  F. 

2.  Required  the  temperature  of  a  mixture  of  3  pounds  of 
ice  at  10°  F.  with  12  pounds  of  water  at  60  F.,  after  the  ice 
melts,  there  being  no  loss  of  heat. 

3.  Required  the  resultant  temperature  of  a  mixture  of  6 
pounds  of  ice  at  20°  F.  and  one  pound  of  steam  at  225°  F. 

Here 
0.504  X  6  X  12  +  864  -f  6  (t  -  32)  =  0.48  X  13  -f-  966  +  (212  -  t). 

4.  Desiring  to  determine  the  approximate  temperature  of 
the  gases  at  the  base  of  a  chimney,  a  mass  of  iron  weighing 
8  pounds  was  placed  in  them,  and  after  remaining  a  con- 
siderable time  was  removed  and  submerged  in  100  pounds 
of  water  at  50°  F.,  when  it  was  found  that  the  temperature 
of  the  water  was  raised  to  55°  F. ;  required  the  temperature 
of  the  gases. 

"We  have,  nearly, 

i  X  8  (t  -  55°)  =  1  X  100  X  5 ; 
.  '.  t  =  617°  F.,  nearly. 

5.  How  many  pounds  of  water  at' 200°  F.  will  be  neces- 
sary to  reduce  one  pound  of  steam  at  212°  F.  to  water,  and 
leave  the  final  mixture  of  water  212°  F. 

6.  Required  the  temperature  of  a  mixture  of  one  pound 
of  ice  at  32°  F.,  one  pound  of  water  at  32°  F.,  and  one  pound 
of  steam  at  212°  F. 

Proceeding  in  the  ordinary  way,  we  have 

144  -f  2  (t  —  32)  =  966  -f-  (212  -  t) ; 

.-.las  366°  F., 

an  absurd  result,  since  the  mixture  would  have  a  higher  temperature  than 
that  of  the  hottest  substance. 


110  IMPERFECT   FLUIDS.  [85.] 

The  proper  explanation  is  —  only  a  part  of  the  steam  was  condensed  pro- 
ducing water  at  212°  F.  By  a  sufficient  pressure,  however,  it  may  all  be 
reduced  to  water. 

7.  In  the  preceding  exercise  how  much  steam  was  con- 
densed forming  water  at  212°  F.  and  how  much  steam  re- 
mained uncondensed  ? 

8.  Required  the  resultant  temperature  of  a  mixture  of  5 
Ibs.  ice  at  28°  F.,  20  Ibs.  H,0  at  50°  F.  and  1  Ib.  steam  at 
212°  F. 

9.  A  mixture  of  *  pounds  of  ice  at  32°  —  £„  x  pounds  of 
water  at  60°  F.  and  s  pounds  of  steam  at  t°  F.,  produces  a 
temperature  of  T°  F.;  required  a?,  and  discuss  the  result. 

We  have 

0.504  »  «,  +  144  i  +  »'  (T  -  32)  =  /(say), 
0.48  *  (t,  -  212°)  +  966  s  +  «(212  -  T)  =  S, 

x(T—QQ)  =  W\ 
S-  I 


Now  assume  that 

8  =  I  ;  S  >  I  and  T  >  60  ; 
8>I,  T  =  60  ;  S  >  I,  T  <  60°  (impossible),  S  <  I,  T  <  60°,  &c. 

10.  How  many  pounds  of  water  at  212°  will  be  necessary 
by  mixing  with  5  pounds  of  alcohol  at  40°  F.  to  just  make 
the  latter  boil  under  a  pressure  of  one  atmosphere  ? 

85.  Total  heat  of  evaporation  is  a  conventional 
term  to  indicate  the  heat  absorbed  in  raising  a  liquid  from 
a  fixed  temperature  in  the  liquid  state  to  the  boiling  point 
and  evaporating  it  at  the  latter  temperature.  It  is  the  sum 
of  the  sensible  and  latent  heats  above  the  fiseed  temperature. 
It  is  also  called  the  total  heat  of  the  vapor,  and  in  reference 
to  water  the  total  heat  of  steam.  To  find  it,  we  use  the  first 
of  equations  (A),  or 


in  which  the  first  term  of  the  second  member  is  to  be  in- 
tegrated from  T^  to  rt,  while  the  second  term  of  that  mem- 


[86.]  EVAPORATIVE   POWER.  Ill 

ber  is  zero  during  that  operation,  and  then  the  value  of  the 
last  term  is  found  while  r^  remains  constant,  and  hence  is 
H^  the  latent  heat  of  evaporation.  The  total  value  of  II 
will  be  the  sum  of  these  values ;  hence,  making  J5TV  =  C, 
the  dynamic  specific  heat,  we  have 

H=  67(r1-ra)  +  //ei.  (93) 

The  lower  fixed  temperature  is  that  of  melting  ice,  unless 
otherwise  specified  ;  hence,  in  English  units 

r,  =  TO 

Tl  =  r0+T-  32°, 
where  T  is  the  temperature  on  Fahrenheit's  scale,  and 

11=  tf(r-32)  +  J7e,  (94) 

and  for  water  C  =  778,  and  substituting  the  value  of  JJe, 
(page  95),  we  have 

H  =  778  [1091.7  +  0.305  (T  -  32)] 
=  849342  +  237  (T  -  32) 
=  841758  +  237  T.  (95) 

By  means  of  this  formula  a  table  may  be  computed  that 
will  give  the  "  total  heat  of  steam"  above  the  melting  point 
of  ice. 

86.  Evaporative  power. — If  the  temperature  at 
which  water  is  fed  to  a  boiler  be  T°  F.,  the  foot-pounds 
of  heat  which  must  be  supplied  in  order  to  evaporate  it 
will  be 

II  =  778  [1091.7  +  0.305  (T  —  32)  —  (T,  -  32)].    (96) 

In  determining  the  efficiency  of  a  boiler,  or  the  amount 
of  water  evaporated  by  a  pound  of  fuel,  it  is  customary  to 
reduce  the  amount  of  evaporation  which  actually  takes 
place  from  the  temperature  of  the  feed  water  at  the  temper- 
ature of  the  steam  to  an  equivalent  amount  at  and  from 
212°  F.  (100°  C.).  In  the  latter  case  966  heat  units  are 
absorbed,  and  making  this  the  unit  of  evaporative  power, 
the  evaporative  power  in  any  other  case  will  be,  nearly, 


112 


IMPERFECT   FLUIDS. 


[86.] 


1092  +  .3  (T  -  32)  -  (T,  -  32) 

966 
0.3  (T  -  212)  -f  (212  -  T,) 

•   966 

a  form  due  to  Rankine,  who  properly  called  the  expression  a 
factor  of  evaporation.  By  assuming  a  series  of  values  for 
T  and  Tv  a  table  may  be  formed  of  the  factors  by  means  of 
which  the  given  conditions  may  readily  be  reduced  to  that 
of  the  above  unit,  and  the  actual  evaporative  power  will  be, 
in  foot-pounds, 

778  X  966  X  tabular  number. 
The  preceding  expression  reduces  to 

,    148.4  +  0.3  T  -  T, 


966 


(97) 


by  means  of  which  the  following  table  has  been  computed. 


FACTORS  OF  EVAPORATION. 


Boiling 


Initial  temperature  of  the  feed  water,  Ty  degrees  F. 


point. 

V°F. 

| 

40 

60 

80 

100 

120 

140 

160 

180 

300 

212 

212 

1.18 

1.15 

1.13 

1.11 

1.09 

1.07 

1.05 

1.03 

1.01 

1.00 

230 

1.181.16 

1.141.12 

1.10 

1.08 

1.061.04 

1.02 

1.01 

250 

1.191.17 

1.15 

1.12 

1.10 

1.08 

1.06 

1.04 

1.02 

1.01 

270 

1.19.1.18 

1.15 

1.12 

1.10 

1.08 

1.06 

1.04 

1.02 

1.02 

290 

1.19,1.19 

1.161.  141.  12il.  10 

1.071.04 

1.03 

1.02 

310 

1.201.20 

1.161.141.1211.10 

1.08 

i  .(•:. 

1.04 

1.03 

330 

1.211.21 

1.17(1.151.13 

1.11 

1.0911.07 

1.05 

1.03 

350 

1.22 

1.21 

1.191.17 

1.15 

1.12 

1.10 

1  .OS 

1.06 

1.04 

370 

1.231.21 

1.19,1.17 

1.15 

1.12 

1.10 

1.08 

1.06 

1.04 

390 

1  .  24  1  .  22 

1.191.17 

1.15 

1.13 

1.11  1.09 

1.07 

1.05 

410 

1.24 

1.22 

1.201.18 

l.lf)  1.14 

1.12 

1.10 

1.08 

1.06 

430 

1.24 

1.22 

1.2111.19 

1.171.15 

1.13 

1.11 

1.09 

1.07 

[87. J  SUPERHEATED    STEAM.  113 

87.  Superheated  steam. — "When  the  temperature 
of  a  vapor  for  a  given  pressure  is  higher  than  the  boiling 
point  for  that  pressure,  the  vapor  is  said  to  be  superheated, 
and  is  sometimes  called  "  steam  gas."  Yapor  may  be  super- 
heated by  separating  it  from  its  liquid  and  subjecting  it  to 
a  still  higher  temperature.  Let  the  vapor  be  generated  at 
tl  degrees  and  afterward  heated  to  r^,  degrees,  then  will  the 
heat  absorbed  above  TI  degrees  be, 

rT<* 

II  =-.  //e  +    /      /rp  d  r  =  II,  +  K^  (r.  -  rt).     (98) 

If  the  vapor  be  steam  generated  from  water  at  Tn  degrees 
Fah.  evaporated  at  'L\  degrees,  and  superheated  at  constant 
pressure  to  T^  degrees, 

H=  778  [(T,  -To )  -f- 1121.7  -  0.695  T,  +  0.48(ra  -  r,)] 

=  778  [1121.7  —  0.175  TI  -  Ta  -f-  0.48  TiJ.  (99) 


EXERCISES. 

1.  If  one  pound  of  coal  will  evaporate  10  pounds  of  water 
at  and  from  212°,  how  many  pounds  would  it  evaporate 
from  80°  F.  at  310°  F.  ? 

2.  If,  when  the  feed  water  is  at  32°  F.  and  the  boiling 
point  at  410°  F.,  one  pound  of  coal  evaporates  7  pounds  of 
water,  how  much  ought  it  to  evaporate  at  and  from  212° 
F.?    - 

3.  Experiment  proves  that  one  pound  of  good  coal,  com- 
pletely burned,  will  develop  14500  heat  units  (B.  T.  U.), 
how  many  pounds  of  water  could  one  pound  of  such  coal 
evaporate  at  and  from  212°  F.  if  all  its  heat  of  combustion 
were  utilized  for  that  purpose,  under  the  pressure  of  one 
atmosphere  ? 

Ans.    15.0  Ibs. 

-1.  Under  what  physical  conditions  could  one  pound  of 


114  IMPERFECT   FLUIDS.  [88. j 

such  coal  as  mentioned  in  Exercise  3  evaporate  20  pounds 
of  water  ? 

5.  If  the  feed  water  be  32°  F.,  what  must  be  the  tempera- 
ture and  pressure  that  the  "  factor  of  evaporation"  shall  be 
2.0? 

6.  Find  the  B.  T.  U.  required  to  produce  one  pound  of 
saturated  steam  at  212°  from  water  at  32°  ;  and  steam  gas 
at  the  same  temperature  from  the  same  water,  and  compare 
the  results. 

(Those  who  have  not  time  to  pursue  the  more  abstruse  part  of  the  sub- 
ject may  omit  to  Art.  98,  p.  143,  except  Art.  95,  p.  126.) 

88.  Free  expansion  of  gases.  When  the  exter- 
nal pressure  is  much  less  than  the  expansive  force  of  the 
gas  during  expansion,  the  expansion  is  said  to  \>efree.  This 
principle  has  been  used  for  determining  the  difference  be- 
tween the  absolute  zero  of  the  perfect  scale  and  that  of  the 
air  thermometer.  When  a  gas  rushes  freely  from  a  vessel 
under  pressure  into  another  of  lower  pressure,  the  only  work 
done  will  be  that  of  the  friction  through  the  passage  and 
among  its  own  particles,  which  process  will  generate  heat ; 
but  during  the  expansion  in  the  second  vessel  the  gas  will  be 
cooled,  and  if  there  were  no  transmission  of  heat  to  or  from 
external  bodies  and  the  gas  were  perfect  the  final  tempera- 
ture should  be  the  same  as  the  initial. 

Joule  made  an  experiment  upon  air,  by  immersing  in  a 
vessel  of  water  two  other  vessels  connected  by  a  pipe,  one 
of  which  was  filled  with  air  at  22  atmospheres  and  the 
other  exhausted  of  air ;  after  which,  by  opening  a  stop- 
cock in  the  connecting  pipe,  the  air  rushed  from  one 
vessel  to  the  other,  but  no  apparent  change  of  temperature 
was  observed  (Phil.  Mag.  (3),  (1845),  XXYL,  376).  M. 
Hirn,  in  1865,  made  a  more  delicate  experiment  for  the 
same  purpose,  but  without  detecting  any  change  in  tempera- 
ture (Tkeorie  Mecanique  de  la  Chaleur,  36me,  I.,  298). 

Sir  William    Thomson  in  1851    executed  a  much  more 


£88.]  FEEE   EXPANSION   OF   GASES.  115 

delicate  experiment.  A  porous  plug,  composed  of  a  bunch 
of  fine  silk  —  or,  in  some  cases,  of  cotton  —  was  inserted  in  a 
long  tube,  and  the  difference  of  pressures  on  either  side 
was  regulated  by  the  amount  of  silk  or  cotton  in  the  plug. 
The  air  was  forced  through  a  box  with  a  perforated  cap 
stuffed  with  cotton-wool,  so  as  to  prevent  fluctuations,  and 
this  pump  was  worked  some  time,  so  as  to  secure  steady 
action  before  records  were  made.  The  pressure  and  tem- 
perature were  kept  nearly  constant  during  the  experiment. 
The  part  of  the  tube  beyond  the  plug  was  immersed  in  a 
vessel  of  water,  observations  upon  which  determined  the 
amount  of  cooling  (Thomson's  Mathematical  Papers,  pp. 
333-455  ;  Phil.  Mag.  (1852),  (4),  IT.,  481).  Since  heat 
will  generally  be  abstracted,  H  will  be  negative,  and  the 
second  of  equations  (A)  becomes 

-dH=  K^dr  -  rj^dp. 
Adding  and  subtracting  v  dp,  we  have 
—  d  II  =  Kv  d  r  —  (r(-  --  v  J  dp  —  v  dp  ; 

dp.     (100) 

considering  Kv  as  constant. 

But  integrating  v  d  p  by  parts,  between  the  limits  of  p,  v9 
and  vl  PV  gives 


v  dp  =  v^p,—  v,p,  +v*P  d  v- 

The  last  term  would  be  negative  if  the  order  of  the  limits 
were  reversed.  But  the  work  done  upon  the  gas  in  forcing 
it  through  the  plug  will  be  nearly  the  external  work  for  the 
sensibly  perfect  gases  ;  or 


and  substituting  these  in  equation  (100),  gives 


116  IMPERFECT   FLUIDS.  [89.] 

f(r  ~  -  *>)  dp  =  R,  (r,  -  T,)  +  v,pt  -  v,p,  (101) 

But  this  cannot  be  reduced,  since  r  depends  upon  the 
zero  of  the  perfect  scale,  which  we  do  not  know  ;  we,  how- 
ever, know  by  experiment  that  it  is  not  far  from  the  zero  of 
the  air  thermometer  ;  hence,  if  t  be  the  temperature  from  the 
absolute  zero  of  the  air  thermometer,  we  have  from  equa- 
tion (2) 

p  v      Po  v0 

-j   -     —j-    -  R,  very  nearly  ; 

dv  _  7? 
'  di  =  1? 
Rt 


dv       d  v 

^  =  jrvei 

r,  —  r,  =  tf,  —  tn  almost  exactly, 

for  the  range  of  temperatures  in  an  experiment,  and  con- 
sidering the  specific  heat  as  constant,  or  J5TP  =  <7P  ;  these 
will  reduce  equation  (101)  to 

~      ~~'  (102) 


which  shows  that  the  absolute  zero  of  the  perfect  scale  is 
lower  than  the  zero  of  the  air  thermometer. 

89.  The  absolute  zero,  found  by  experiments  upon 
air,  oxygen  and  nitrogen,  by  the  aid  of  equation  (102),  has 
a  mean  value  of 

492.66°  F.  =  273.7°  C. 

below  the  melting  point  of  ice  (Thomson's  Papers,  p.  392), 
while  the  zero  of  the  air  thermometer  is  491.13°  F.  below. 
so  that  the  absolute  zero  is  about  1.53°  F.  below  that  of  the 


[90,  91.]  SPECIFIC   HEAT.  117 

air  thermometer.  The  difference  is  so  small  compared  with 
the  distance  from  the  melting  point  of  ice  as  to  render  it 
probable  that  the  approximation  is  very  close,  supposed  to 
be  within  --  of  the  exact  value. 


SPECIFIC  HEAT. 

9O.    General    expression    for    specific    heat. 

The  first  of  equations  (A)  gives 


which  is  a  general  expression  for  the  sp.  heat  of  any  sub- 
stance at  the  volume  v  and  temperature  T,  the  path  of  the 
fluid  being  arbitrary.  Unless  otherwise  stated,  the  path  is 
assumed  to  be  either  parallel  to,  or  perpendicular  to,  the 
-y-axis,  giving  rise  to  the  conditions  j?  constant  or  v  constant  ; 
hence,  dp  =  o  or  d  v  —  0,  which  in  equations  (A)  give  the 
respective  equations 


as  previously  established  in  Articles  38  and  39. 

91.  Specific  heat  at  a  change  of  state  of 
aggregation.  It  has  been  shown  that  for  fusion,  evapo- 
ration and  sublimation  the  change  of  state  takes  place  at  a 
fixed  temperature  corresponding  to  a  given  pressure  for 
each  substance  ;  hence,  for  these  cases  we  have 
d-c  =  o,  d  H  >  0; 


similarly, 

JTT=  Jr,-J?=±co; 

or  the  specific  heats  AT  the  change  of  state  are  infinite, 


118 


IMPERFECT  FLUIDS. 


[92.] 


92.  Modified  expression  for  the  specific  heat. 

The  specific  heat  of  substances  which  are  capable  of  a  large 
expansion— conceived  to  be  indefinitely  large — admit,  ac- 
cording to  the  theoretical  views  of  Rankine,  of  being  ex- 
pressed in  two  parts — a  constant  and 
a  variable  part.  To  show  this,  let  b, 
Fig.  34,  be  the  initial  state  of  one 
pound  of  the  substance,  b  e  an  isother- 
mal for  the  substance,  c  d  a  consecutive 
isothermal,  Kv  the  specific  heat  at  the 
volume  v,  Kv-  the  specific  heat  at  the 
volume  v".  From  the  state  b  let  the 
temperature  be  increased  d  r,  thus  in- 
creasing the  pressure  an  amount  b  c ;  from  the  state  c  ex- 
pand it.  doing  the  external  work  v  c  d  v",  at  d  abstract  heat, 
so  as  to  reduce  the  temperature  d  r,  reducing  the  pressure 
to  v"  e,  thence  compress  it '  isothermally  to  J,  doing  work 
i)"  e  b  v  upon  the  substance,  then  will  the  resultant  external 
work  be 

f*v" 

Icde  —    I     dpdv, 
Jv 

and,  according  to  the  second  law,  omitting  the  parentheses 
indicating  partial  differentials, 


^        -V 
FIG.  34. 


/v 


Also, 


m,  5  c  mt  =.  J£"v  d  r, 
=  K^-  dr; 


[92.]  SPECIFIC   HEAT.  119 

And, 

m,  b c m,  -j-  m3  c  d m6  —  m,  b  em4  —  mtedms  —  b  cde\ 

hence,  substituting  and  reducing, 


Removing  v"  indefinitely  to  the  right,  we  have  v"  =  <x> , 
and  then  JTV.  becomes  K^ ,  which,  according  to  a  theory  of 
Rankine,  is  constant  for  a  constant  state  of  aggregation, 
according  to  which  the  preceding  equation  becomes 

v,  (105) 

in  which  the  last  term  is  the  rate  at  which  the  internal  work 
is  done  at  the  volume  v  and  temperature  r,  due  to  a  change 
of  temperature.  It  may  be  found  more  directly  from  equa- 
tion (67)  by  differentiating  it,  considering  r  as  the  independ- 
ent variable,  giving 


and  dividing  by  d  r  gives  the  heat  doing  internal  work  per 
unit  of  temperature,  which  is  the  same  as  that  above  when 
the  corresponding  limits  are  assigned.  Equation  (105)  being 
the  specific  heat  at  the  volume  v  and  temperature  r,  we 
have,  for  the  heat  absorbed  at  constant  volume, 

'  JT      J        d'** 

the  last  term  of  which  may  be  integrated  when  the  equa- 
tion of  the  gas  is  given. 

Similarly,  if  p  and  r  be  the  independent  variables,   we 


120  IMPERFECT   FLUIDS.  [93.] 

would  find 

K9  =  C  +  R  -  r  /*(££)  dp,  (107) 

Jo 

and  for  an  increase  of  temperature  r,  —  r^  the  heat  absorbed 
at  constant  pressure  would  be 


H=  f  KvdT= 
J 


93.  The  apparent  specific  heat  is  the  total  heat 
absorbed  by  unity  of  weight  in  producing  an  increase  of 
temperature  one  degree,  and  includes  that  necessary  to 
make  the  substance  hot  as  well  as  that  doing  the  internal 
work,  and  also  the  external  in  the  case  of  constant  pressure. 
It  is  represented,  for  expansible  gases,  by  the  second  mem- 
bers of  equations  (105)  and  (107). 

The  real  specific  heat  is  that  part  of  the  apparent  specific 
heat  which  directly  makes  the  substance  hotter.  It  is  rep- 
resented by  C  in  equation  (105),  and  is  called  actual 
energy. 

The  apparent  specific  heat  is  the  sum  of  the  actual  and 
potential  energies  in  the  particular  specific  heat.  Equations 
(105)  and  (107)  are  the  dynamic  specific  heats,  and  to  find 
them  in  ordinary  units  they  must  be  divided  by  J. 

The  theory  of  Rankine,  referred  to  above,  is  "  The  real 
specific  heat  of  each  substance  is  constant  at  all  densities  so 
long  as  the  substance  retains  its  condition,  solid,  liquid,  or 
gaseous"  (Prime  Movers,  p.  307).  The  correctness  of  this 
theory  has  been  questioned  by  Clausius,  and  in  one  place 
Kankine  says  "it  is  prolatdy  constant"  (ibid.,  p.  250).  The 
theory  is  useful  in  showing  the  different  effects  probably 
produced  by  the  absorbed  heat  ;  but  it  is  of  no  service  ex-. 
cept  in  expansible  gases,  and  cannot  be  used  in  those  cases 
except  where  the  equation  of  the  fluid  is  known  —  and  it  is 
known  for  only  a  few  substances. 


[93.]  SPECIFIC   HEAT.  121 

EXERCISES. 
1.  Find  the  specific  heat  at  volume  v  and  temperature  r  for 

the  gases  represented  by  the  equation^?  v  =  R  t 

We  have 


,    a  a 

+       f 


dv       2  a 


From  this  result  it  appears  that  the  specific  heat  at  constant  volume 
decreases  as  the  temperature  increases,  for  all  gases  represented  by  the 
above  equation,  and  approaches  C  as  a  limit. 

2.  Find  the  heat  necessary  to  raise  the  temperature  of  one 
pound  of  the  gas  of  the  preceding  exercise  from  rl  to  rt  at 
the  constant  volume  v. 

Equation  (106)  becomes 


From  this  it  appears  that  G  -\ is  the  mean  specific  heat  between  r> 

"0  ^aTi 
and  r,. 

S.  Find  the  heat  absorbed  by  one  pound  of  COt  in  rais 


122  IMPERFECT   FLUIDS.  [94.J 

ing  its  temperature  from  500°  F.  to  600°  F.  at  the  volume 
8.5  cubic  feet. 

From  equation  (6),  page  13,  we  have 

R  =  35,         a  =  481600, 
and  the  preceding  equation  gives  (C  being  132), 


/ 
H  =  (132  +       ~)  100  =  (132  +  0.377)  100 

=  13238  foot-pounds. 

It  will  here  be  observed  that  the  term  due  to  internal  work  is  very  small 
compared  with  the  actual  energy,  and  may  properly  be  omitted,  especially 
when  we  consider  ia  addition  thereto  that  it  is  less  than  the  errors  of  ob- 
servation determining  the  number  132. 

4.  Find  the  apparent  specific  heat  of  the  gases  represented 
by  the  equation  p  v  =  R  r.  (Use  equations  (105)  and 
(107).) 

94.  General  expression  for  the  difference  of 
specific  heats. 

In  equation  (103)  the  left  member  is  the  specific  heat  at 
constant  pressure,  if  p  be  constant  ;  hence, 


To  make  this  more  apparent,  use  both  of  equations 
which  are 

dv, 


subtracting,  gives 


In  Figure  35  let  A  r  and  B  C  be  two  consecutive  isother- 
inals  and  A  1>  the  path  of  the  fluid,  A  B  perpendicular  to  the 


L94.1 


SPECIFIC    HEAT. 


123 


axis  of  volumes,  A  £7  parallel  thereto,  then,  as  shown  by  equa- 
tion (20),  will 


r  (  — ±- 1  d  v  =  m.  B  b  m, : 

d  rl 


in  which  d  v  in  the  former  expression 

is  the  abscissa  of  b  from  A  =  A  a, 

and  dp  =  a  b  in  the  latter  is  the  or-  FIG   3g 

dinate  of  b  from  A  C.     If  d  v  in  the 

former  be  made  the  abscissa  of  C  in  reference  to  A,  that  is, 

d  v  =  A  C.  then  would  T  (  -JL  \  d  v  be  the  area  m^  B  C  m4, 
\a  rJ 

which  is  the  sum  of  the  two  preceding  expressions,  anc1 
hence  the  value  of  the  expression  in  the  [  ]  of  the  former 
equation.  To  indicate  that  d  v  extends  thus  far,  it  is  nec- 
essary to  express  v  as  a  function  of  r,  since  it  will  be  limited 
by  two  isothermals,  and  will  be  written  thus  : 


then 


which,  substituted,  gives 

-=, 


\drJ  \<lrJ' 
as  before. 

In  soUds  and  liquids  it  is  difficult  to  find  the  variation  of 

pressure  with  temperature  at  constant  volume,  f-r^-J  ,  and 
that  factor  may  be  eliminated  as  follows : — 


(d  v  \  (d  v  \ 

v  =  [  -7—  )  d  p  4-  (  -,—  }  d  r. 

\dp)     ^~  \d-rj 


124  IMPERFECT   FLUIDS.  [94.] 

But  for  v  constant  d  v  =  o ; 


which  in  (109)  gives 

(^M2 

(110) 


an  expression  equally  true  for  gases. 

EXERCISES. 

1.  Find  the  difference  between  the  specific  heats  of  a  per- 
fect gas,  the  equation  being  p  v  =  It  r.     (Equation  (109) 
or  (110).) 

2.  Find  the  difference  between  the  specific  heats  of  a  gas 

whose  equation  isp  v  =  R  T  — 

TV 

v  f      ,   2  a\  /TVV*  -\-  2«> 
Ans.    Ap  —  Av  =  -  \p-\ » 


3.  Find  the  difference  between  the  specific  heats  of  car- 
bonic acid  gas  at  the  melting  point  of  ice  at  the  pressure  of 
one  atmosphere. 

4.  Find  the  difference  between  the  specific  heat  of  water 
at  constant  pressure  and   at  constant  volume  when  at  its 
maximum  density. 

It  is  at  its  maximum  density  under  the  pressure  of  one  atmosphere  at 
the  temperature  of  39.1°  F.,  and  at  this  state 

ldv\   _ 


5.  Find  the  difference  between  the  specific  heats  of  water 
at  39°  F.  and  77°  F.  under  the  pressure  of  one  atmosphere, 


[94.J  SPECIFIC   HEAT.  125 

the  coefficient  of  expansion  at  77°  F.  being  0.00014  of  its 
volume  per  degree,  and  the  coefficient  of  compression  being 
0.000046  of  its  volume  for  one  atmosphere. 

The  volume  of  one  pound  of  water  at  maximum  density  being  0.016 
cubic  feet,  we  have 

*=  ^016X0.00014).  =  A,- 


Assuming  that  the  rate  of  expansion  for  38°  (=  77  —  39)  is  uniform, 
the  volume  at  77°  F.  will  be 

0.016  -}-  0.016  X  0.00014  X  33  —  0.01609,  nearly  ; 
hence,  for  a  pressure  of  one  pound  per  square  foot  we  have 
X  0.000046  351 


_ 
dp)  ~  2116.2  1013 

'  .  K  -  Kv  =  537.6  X  aihr  =  7.68  ; 


A  similar  computation  at  122°  gives 


Adopting  Regnault's  values  for  the  specific  heat  at  constant  pressure, 
we  find 

i  c  =1.0000 

at39 1  ;= 1.0000 

at  77°  j  C?  =  1'0016 
7    \cv  =  0.9917 

at  122° 

It  will  be  observed  that  while  the  value  of  cp  increases  with  the  temper- 
ature cv  decreases,  and  hence  the  difference  increases  more  and  more  as 
the  temperature  increases.  The  exact  numerical  results  here  found  are 
not  to  be  relied  upon,  but  they  are  approximately  correct,  and  indicative 
of  a  general  law. 

6.  Show,  both    analytically  and  geometrically,  that  the 
specific  heat  for  constant  pressure  exceeds  that  at  constant 
volume. 

7.  Show  that  the  term  r  i  -—^  d  v  disappears  for  gases 
whose  equations  (if  any)  are  p  v  =  R  r  -f-  cp  (v). 


126  IMPERFECT   FLUIDS.  [95,  96.] 

95.  Water  is  the  only  substance  whose  specific  heat  has 
been  accurately  found  for  a  large  range  of  temperatures. 
According  to  Regnault's  experiments,  there  is  a  small  increase 
in  the  -specific  heat  with  every  increase  of  temperature  above 
32°  F.,  but  according  to  the  very  accurate  experiments  of 
Professor  Rowland,  the  specific  heat  of  water  decreases  from 
4°  C.  to  27°  C.,  and  may  be  approximately  represented  by 
the  empirical  formula 

c=l-  0.00052  (t  -  4)  +  0.000003  (t  -  4)',     (111) 

in  which  t  should  not  exceed  30°  C. 

Rankine's  formula  for  representing  Regnault's  experi- 
ments is 

c  =  1  +  0.000000309  (T  -  39.1)"  (Fall.),       (112) 
=  1  +  0.000001  (t  -  4)'*  (Cent.). 

M.  Bosscha  represents  Regnault's  experiments  by  the 
formula 

c  =  1  +  0.00022  t  (Cent).  (113) 

When  the  law  of  the  specific  heat  is  known,  the  number 
of  thermal  units  absorbed  in  raising  a  pound  of  the  sub- 
stance from  T,  to  Ty  degrees  will  be 

h=    f   *cdT, 


-I 
JTV 


and  the  mean  specific  heat  will  be 
h 

-*•  2  •*•  \ 

96.  Another  form  of  the  general  equation. 

Substituting  the  value  of  A'v,  equation  (105)  in  the  first  of 
equations  (A)  gives 


r  I      ^drdv 

J  00 


dH=  C  dr  +  r  drdv  +  rdv.  (114) 


*  Trans.  R.  S.  E.,  XX.  (1851),  441. 


[96.] 


GENERAL   EQUATION. 


127 


This  is  analyzed  by  Rankiue  as  follows  (Prime  Movers,  p.  312)  :— 

"  I.  The  variation  of  the  actual  heat  of  unity  of  weight  of  the  fluid 
Cvdr. 

"  II.  The  heat  which  disappears  in  producing  work  by  mutual  molec- 
ular actions  depending  on  change  of  temperature  and  not  on  change  of 
volume. 

"  III.  The  latent  heat  of  expansion,  T  —~  d  r,  that  is,  heat  which  dis- 
appears in  performing1  work,  partly  by  the  forcible  enlargement  of  the 
vessel  containing  the  fluid  and  partly  by  mutual  molecular  actions  de- 
pending on  expansion." 

The  integral  of  the  last  equation  would  if  determined, 
give  the  heat  absorbed,  in  foot-pounds,  in  producing  the 
changes  of  temperature  and  volume  ;  but  the  integral  can- 
not be  performed  without  the  equation  of  the  path  of  the 
fluid  and  the  equation  of  the  fluid. 

Still  another  form.  Subtracting  pdv,  the  external 
work  done,  from  both  members  of  the  preceding  equation 
gives 


-£-p  rf«,(115) 


in  which  each  member  is  the  excess 
of  the  heat  absorbed  above  the  exter- 
nal work  done.  The  several  terms 
may  be  represented  by  Fig.  36,  in 
which  A  B  is  the  path  of  the  fluid, 
A  C  an  isothermal  through  A,  and 
A  a  c  C  represents  the  internal  work 
done  during  the  isothermal  expansion 
A  C.  Then 

H  =  ml  A  B  m3, 

fp  d  v  =  v,  A  B  v.,  =  v,  A  Cvt  + 


C. 


I    I  T  —£.  d  r  dv  =  areas  represented  ~by  the  dotted 
J  J     d  T 

between  C  m^  and  B  m3, 


128 


IMPERFECT    FLUIDS. 


[96] 


C  (T,  —  r,)  =  m,  C  B  m3  —  areas  represented  by  the  dot- 
ted lines, 

C  (r,  —  *,)  +    /   ft  ~^  d  T  dv  =  m,  C  B  ms, 

«/     »/  (IT 

v  =  A  a  c  C, 


vi  a  c  vt  =  ml  A  C  my  ; 
H  —  ip  d  v  —  m,  A  B  m3  —  vt  A  B  vt 


It  is  apparent  that  the  second  term  of  the  second  member 
of  equation  (115)  is  the  <p  (r)  of  equa- 
tion (67). 

The  internal  work  may  be  repre- 
sented in  another  form.  In  Fig.  37 
let  A  r,  and  B  r,  represent  isother- 
mals  extended  indefinitely,  and  con- 
ceive that  rt  A  B  rt  forms  a  closed 
cycle,  then  will  the  resultant  internal 
work  in  passing  around  the  cycle  be 
zero.  Let  a#b  be  the  internal  work  done  in  passing  from  A 
to  B)  v#»  in  passing  from  B  to  infinity,  ^  in  passing 
from  infinity  back  to  A  ;  then 

A  +  A-  A  =  o.  (no) 

Let  n  be  the  isothermal  through  A,  T»  through  B,  Figs. 
36  and  37,  then,  equation  (115), 


FIG.  37. 


and  from  Fig.  37,  $a  being  the  potential  energy  at  A,  etc. 

(118) 

(119) 

*/oo 


[96.]  GEtfEKAL   EQUATION.  129 

these  combined  with  equation  (116)  reduces  (115)  to 

//  —      p  dv  =  (C'r  —  £)„  -  (C  r  -  &)a,      (120) 


in  which  (C  r  —  S\  is  called  the  intrinsic  energy  of  the 
gas  in  the  state  B,  and  depends  upon  the  state  of  the  gas 
only ;  hence,  the  heat  absorbed  above  that  necessary  to  per- 
form the  external  work  equals  the  increase  of  the  intrinsic 
energy. 

EXERCISES. 

1.  If  the  equation  of  the  gas  bej>  v  —  R  r  —  — ,  verify 

the  statement  that  the  internal  work  is  the  same  whether 
the  path  be  A  C  and  C  J2,  Fig.  36,  or  the  indefinitely  ex- 
tended isothermals  ra  and  rb.  (Equations  (117),  (118),  (119).) 

2.  If  the  equation  be  the  general  one  given  by  Kankine, 

p  v  =  12  T  —  a0  —  — ~  —  &c.,  a.,,  a.,  a.,  &c.,  being 

TV  T2  V 

constant,  verify  the  fact  that  the  internal  work  is  the  same, 
whether  A  C  and  C  B  be  the  path,  or  whether  the  path  be 
along  two  isothermals  through  A  and  £,  respectively,  in- 
definitely extended. 

3.  Test,  the  same  principle  for  the  ideal  gas  whose  equa- 
tion is  p  V*  =  c  T,  c  being  a  constant. 

4.  Will  the   principle  stated  in  Exercise  2  be  true  if  the 
equation  of  the  gas  were  p  =  v  T  ? 

5.  Find  the  internal  work  done  by  ex- 
pansion at  constant  pressure  from  vl  to  v, 

when  the  equation  of  the  gas  i&pv  =         n          ^] f 

R  r  _  ^L.    ( (117),  or  (118)  and  (119).) 


Ans.    2  a\— L.T 

LT,  vl       r9  vy_\ 


V,  1/4 

FIG.    38. 


This  is  the  value  of  the  last  term  of  equation  (108).     The 
last  term  of  equation  (108)  is  not  easily  found  directly,  since 


130  IMPERFECT   FLUIDS.  [96.] 

it  contains  internal  work  due  botli  to  a  change  of  volume 
and  temperature.  Since  r2  increases  with  i\  when  p  is  con- 
stant, it  follows  from  the  preceding  Answer  that  the  internal 

work    increases   with  the   expansion  and    approaches    the 

f)   . 
limit  _-  —  ,  which  is  a  function  of  the  initial  state  only,  and 

TI  Vl 

decreases  with  the  increase  of  temperature.  It  may  be  in- 
ferred from  this  —  although  we  have  by  a  more  complete 
analysis  found  —  that  the  last  term  of  equation  (107)  is  not 
only  essentially  negative  but  decreases,  numerically,  with  in- 
crease of  temperature  ;  hence,  the  internal  work  of  all  ex- 
pansible substances  whose  equation  approximates  to  the  fonn 

p  v  =  R  T  --  increases  with  increase  of  temperature  un- 

der constant  pressure. 

6.  Find  the  real  dynamic  specific  heat  of  carbonic  acid 
gas  at  the  pressure  of  one  atmosphere. 

We  have 

<7P  =  CV  +  R  =  132  +  =  167  foot-pounds. 


7.  Find  the  internal  work  of  expanding  C  O^  from  v,  = 
8.5  cu.  ft.  at  rl  —  500°  F.  to  T,  =  600°  F.  at  a  constant 
pressure. 
From  equation  (6)  we  have 

5500__481600__=20471b 
8.5       500  X  (8.5)* 

_35_XJ>00       481600 
vt  600  «,»  ' 

.  •  .  r>i  =  10.22  cubic  feet. 
Substituting  in  the  Answer  of  Exercise  5,  gives 

2  X  481600  [500  x  a5  ~  eoo  X1  10.22]  =  70  ^-pounds,  nearly. 

But  this  is  the  work  for  an  increase  of  100  degrees  of  temperature  ; 
hence,  the  average  will  be  0.70  foot-pounds  per  degree  of  temperature. 
This  is  less  than  ^ff  of  the  heat  producing  actual  energy  of  ihe  substance 


[97.]  GENERAL   EQUATIONS.  131 

per  degree  of  temperature,  as  will  be  found  by  comparing  this  result  with 
that  given  in  the  preceding  Exercise. 

From  the  preceding  analysis  it  appears  that  for  ordinary 
engineering  purposes  the  specific  heat  of  all  substances  may 
be  considered  constant  for  a  constant  state  of  aggregation  ; 
and  the  most  important  element  involving  the  imperfection 
of  the  fluid  is  that  due  to  a  change  of  state  of  aggregation. 
It,  however,  furnishes  a  wide  field  for  scientific  investigation. 

97.  Other  forms  of  the  general  equations. 
In  Fig.  39  let  the  path  A  b  be  intersected  by  equidistant  iso- 
thermals,  of  which  T  through  A  and 
r  -f-  d  r  through  b  are  consecutive. 
Through  A  draw  the  horizontal  A  a 
and  through  b  the  vertical  b  a ;  then 
will  the  heat  absorbed  in  passing  from 
A  to  a  at  constant  pressure  be 


excepting  that  d  r  is  not  independent,  FIG  39 

but  is  dependent  upon  d  v,  the  abscissa 
of  b  in  reference  to  A.  and  hence  we  have 


Similarly  the  heat  absorbed  from  a  to  1  at  constant  vol- 
ume will  be 


hence,  ultimately,  the  heat  absorbed  in  working  from  A  to 
will  be  the  sum  of  these,  or 


a  form  given  by  Zeuner  (  Theorie  Mecanique  de  la  CkaZeur, 
p.  547)-. 

Substituting  {-\  found  from  equation  (109)  and  (-3-^ 


132  IMPERFECT   FLUIDS.  [97.] 

from  the  same,  reduces  this  to 


which  may  be  found  directly  from  equations  ( A)  by  elimi- 
nating d  T  between  them.  This  form  is  given  by  Clausius 
(on  Heat,  p.  179). 

Again,  from  equation  (A}t  we  have,  for  r  constant, 


and  from  equation  (105) 


L\      *L  (<LK\  -  (d  p\  -  1  (d  H\ 

d*\dv)r  ~  d~v  \~d^)v  ~\dr)  -r\dvr 
in  which  r  and  v  are  the  independent  variables.  Similarly, 
from  equation  (A\  and  (107),  we  find 


d    (d  H\         d    (dH\_        (d  v\  . 
T~r  \dp),~~  Jp  \dVJP=   ~  \d~iJ  ' 

in  which  T  and  p  are  the  independent  variables.  These 
forms  are  given  by  Clausius,  though  obtained  in  a  very  differ- 
ent manner  (Mechanical  Theory  of  Heat,  Clausius  (Browne's 
translation),  pp.  118,  119). 

Again,  from  equation  (109),  we  have 


[97.] 


GENERAL   EQUATIONS. 


133 


1  -  i  (rf?)  (rf ) 


a  form  used  by  Professor  Kowland  in  his  paper  on  the  Me- 
chanical Equivalent  of  Heat,  page  146. 

Again,  in  Fig.  40  let  rl  be  the  temperature  of  the  iso- 
thermal A  B,  rt  that  of  j  i,  d  H,  the 
heat  absorbed  at  rl  between  the  con- 
secutive adiabatics  A  (p,  and  b  n  ;  then, 
according  to  Article  40,  equation  (20), 
we  have 


i>.  vv3v*     v+ 
FIG.  40. 


from  which,  the  second  members  being  the  same,  we  have 
dff,       d.II, 


Similarly,  if  r3  be  the  temperature  of  the  isothermal  d  c, 
r.  of  v  z,  we  would  have 


clH,      dff,      dII3      dH._ 

Tl  T2  T3  T4 

or,  considering  heat  emitted  as  essentially  negative,  all  the 
terms  may  be  written  with  the  plus  sign,  and,  generally, 
when  a  succession  of  operations  are  performed  in  Carnofs 
cycle,  we  have 

dH 


or,  ultimately, 


-0. 


134  IMPERFECT   FLUIDS.  [97.] 

But  any  cycle  may  be  divided  into  an  indefinite  number 
of  strips  by  adiabatics  drawn  across  it,  and  by  drawing  iso- 
thermals  from  their  intersections  with 
the  path  to  adjacent  adiabatics,  an  in- 
scribed polygon  may  be  constructed 
whose  area  may  be  made  to  differ  from 
that  of  the  given  cycle  by  less  than 
any  assignable  quantity ;  hence,  ulti- 
mately, if  the  integral  extend  through- 


FIG.  41.  out  the  entire  cycle, 

=  0.  (122) 


fd-f 


If  the  integral  be  separated  into  two  parts,  one  along 
A  a  B,  during  which  heat  is  absorbed,  the  other  along  the 
B  J  A,  during  which  heat  is  emitted,  we  have 


rs*n   fAa_B=0 

JA     r     Jn     T 


Equation  (122)  is  Thomson's  generalization  of  the  second 
law.  It  was  first  published  by  Clausius  in  1854  (Poyy. 
Ann.,  Vol.  XCIIL,  p.  500  ;  Clausius  on  Heat,  p.  90). 

(The  differential  of  a  function  of  two  or  more  independent 
variables  is  said  to  be  an  exact  differential.  Let 

Mdx+JTdy 

be  such  an  expression,  in  which  M  and  TV  may  be  functions 
of  a?  and  y  ;  then  it  is  shown  by  the  calculus  that  the  differ- 
ential of  M  in  regard  to  y  equals  the  differential  of  N  in 
regard  to  x,  or,  dM  ^d_N_ 

d  y        dx 

This  principle  has  been  successfully  applied  to  many  prob- 
lems on  heat,  and  of  the  early  investigators,  Thomson  led 
in  this  mode  of  analysis.) 

Since  it  has  been  shown  that  the  integral  of  -    -  is  zero 


[97.]  GENEEAL   EQUATIONS.  135 

for  a  complete  cycle,  it  is  an  exact  differential,  and  may  be 
represented  by  a  single  symbol,  as  q>,  and  we  have 

—  =  ^j 

hence,  from  (^)j  we  have 

-2-^—K-  —       (^}d  (123) 

t  v   T         \ dt ) 

Applying  the  preceding  principle,  we  have 

d_  /Kv\  _   d_  (dP\ 
dv\r)~dr   \d  T/ 
or 

,  ^  (d*p\    . 

d  JKV  =  T  \~j3j  dv, 

which  is  the  differential  of  equation  (105).  In  differen- 
tiating the  left  member,  r  is  not  a  function  of  v,  since  v  is 
constant  during  the  change  of  temperature. 

Finally,  let  d  =  the  internal  energy  of  the  substance 
both  actual  and  potential, 

L  =  the  latent  heat  of  expansion  as  a  thermal  capacity, 
and  other  notation  as  previously  given,  then 

dff=  d E-\-p  dv  =  KT,  d  r  -\-Ldv ;         (124) 

.  *.<^-£7=  Kv  d  r  -j-  (L — p)dv.  (125) 

But  when  any  substance  is  worked  in  a  complete  cycle 

the  resultant  internal  work  is  zero ;  hence,  dE  is  an  exact 

differential,  and  we  have,  omitting  the  parentheses  of  the 

partial  differentials, 

d 


__ 
*   dv        dr       dr 

Similarly,  for  r  and  p  independent  variables 


136  IMPEKFECT   FLUIDS.  [97a."j 

Since is  an  exact  differential,  therefore  will  its  value 

V     7  I       -"    j  f-\  OCX 

—  dr-\ dv  v1^") 

also  be  an  exact  differential ;  hence, 

d.Xv=d.£.  (129) 

dv     T       dr    T> 

dv       d  T       T 

observing  that  r  in  the  left  member  is  not  a  function  of  v  ; 
hence,  d  T  =  0  in  that  member,  and  by  (126) 

dp  =  dL_dJCI  =  L  (13()) 

9 7 a.  The  Thermodyiiamic  Function,  or  En- 
tropy. The  function  tp  Kankine  calls  the  therinodynamic 
function.  The  differential  of  q>,  or  d  (p,  is  the  heat  ab- 
sorbed for  each  degree  of  absolute  temperature  between  zero 

P 

"VB 


FIG.   42. 

and  T,  while  the  substance  is  worked  along  any  path  from 
a  point  on  one  adidbatic  to  a  point  on  the  adjacent  adiabatic. 

First  let  the  path  be  an  isothermal,  as  A  J2,  Fig.  42, 
whose  temperature  is  T. 

Intersect  the  path  by  an  indefinite  number  of  ordinates 
having  jetween  them  the  constant  distance  dv,  and  from 


[97a.]  THE   THEKMODYJSTAMIC   FUNCTION.  137 

the  points  of  intersection  #,  &,  a',  I',  &c.,  draw  adiabatics; 
and  across  tliese  draw  the  isothermals  K  D,  J  f]  G  H,  &c., 
the  successive  ones  differing  by  one  degree,  in  which  case 
the  temperature  of  C  D  will  be  t  —  1,  of  E  F,  T  —  2,  &c. 
Then  will  d  <p,  for  an  expansion  from  state  a  to  state  £»,  be 

d  cp  =  c  dfe\ 

for  we  have  previously  shown  (Article  40)  that  the  heat  ab- 
sorbed in  working  from  a  to  5  isothermally  will  be 

d  II  '  =  y  al)  z  =  r  (-y^-J  d  v  ', 

dH      tdp\  7 

.  •  .  ---  =    -r^-  }  d  v  • 

r         \dr  1 

but  it  was  shown  in  the  same  article  that  isothermals  equi- 
distant in  temperature  divide  the  heat  into  equal  parts  ; 
hence,  cdfe  in  the  figure  will  represent  one  of  those  equal 
parts  ; 


Also, 

dq>  =  abdc  =  c  df  e  =  ef  h  g,  &c. 

And  for  an  expansion  from  state  A  to  state  «,  we  have 

d  y  =  A  a  c  C  =  CceE—EegG,  &c., 
and  similarly  for  other  expansions.     But  C  c  e  E  does  not 
equal  c  d  f  e,  &c.     For  the  sensibly  perfect  gases  equation 
(131)  becomes 

d  cp  =  —  d  v, 

according  to  which  d  <p  diminishes  as  v  increases,  R  and 
d  v  being  constant  ;  hence 

Cce  E>  cdfe  >  d  c'  e'  f,  &c.} 

and  for  v  indefinitely  large  the  area  representing  d  <p  van- 
ishes; 'and  f  or  v  zero,  d  q>  will  be  indefinitely  large.     The 
indefinite  integral  of  the  preceding  equation  is 
q>  =  R  log  v  -\-  C, 


138        .  IMPERFECT  FLUIDS.  [97a.'a 

and  for  v  =  1,  we  have 

g)  =  C  —  (p,  (say)  ; 
.  •  .  <p  —  <p^  =7?  log  v. 

But  the  value  of  <pn  or  (7,  is  unknown,  and  it  is  imprac- 
ticable to  find  the  value  of  <p  ;  a  condition,  however,  which 
is  of  little  consequence,  since  it  is  only  the  difference  of  the 
thermodynamic  functions  that  is  of  any  importance.  Al- 
though this  illustration  is  for  perfect  gases,  yet  the  principle 
is  the  same  for  all  substances.  This  principle  may  be  par- 
tially illustrated  by  our  notation  for  indicating  heat  ab- 
sorbed during  isothermal  expansion.  Thus,  in  Fig.  42,  let 
JTA  be  the  heat  absorbed  in  expanding  from  some  unknown 
state  L  on  the  isothermal  A  B  to  state  A,  and  IIB  the  heat 
absorbed  in  expanding  isothermally  from  the  same  unknown 
state  to  B  ;  then 

7/B—  7/A  =  A//B, 

will  be  the  heat  absorbed  in  expanding  isothermally  from 
AtoJB.  A  similar  notation  will  apply  to  expansion  accord- 
ing to  any  law,  the  points  Z,  A  and  JB  being  on  the  same 
path.  Similarly,  whatever  be  the  unknown  initial  value  of 
<p,  we  may  write 

<P*  —  <?A  =  A<?B. 

The  sum  of  all  the  areas  C  e,  c  f,  &c.,  between  C  E  and 
D  7^  represents  the  heat  absorbed  for  one  degree  of  abso- 
lute temperature  for  an  expansion  from  state  A  to  state  B. 
From  equation  (131),  if  TI  be  the  constant  temperature  of 
A  B,  we  have  by  integration 


=  ABDC,  &c. 


But  it  is  customary  to  compute  the  heat  absorbed  from  the 
initial  state,  A,  in  which  case  77A  =  0,  and  we  have 


[97a.]  THE   THEKMODYNAMIC   FUNCTION.  139 

The  value  of  the  thermodynamic  function  for  £  will  be 

cpB  =  LBDM, 
and  for  J.,  cpA  =  L  A  CM  ; 


Thus  far  we  have  considered  v  as  the  independent  vari- 
able, and  deduced  and  interpreted  d  <p  under  that  hypothesis. 
Now  we  will  make  (p  the  independent  variable,  in  which 
case  d  (p  will  be  constant.  This  condition  will  be  accom- 
plished by  drawing  the  successive  adiabatics  A  cp^  a  e,  5  fy 
&c.,  in  such  a  manner  that  the  areas  A  a  c  C,  a  b  d  c,  &c., 
shall  be  equal,  in  which  case  we  not  only  have 

AacC=  CceE  =  Eeg  G,  &c., 
but  also 

A  a  G  C  =  a  ~b  d  c  =  ~b  a!  c'  d  =  c  df  e,  &c., 

so  that  if  the  entire  space,  <p^AB  $>„  be  divided  by  iso- 
thermals  of  which  the  successive  ones  shall  differ  by  one 
degree  of  temperature,  and  adiabatics  drawn  in  the  manner 
just  described,  all  the  small  areas  thus  formed  will  ~be 
equal. 

Since  Fig.  42  is  used  to  illustrate  two  modes  of  divi- 
sion, it  should  be  observed  that  the  two  sets  of  adiabatics 
will  not  coincide,  but  that  there  will  be  a  less  number  be- 
tween the  initial  and  terminal  adiabatics,  A  (p,  and  B  <pv  in 
the  latter  mode  of  division,  if  the  area  A  a  o  C  be  the  same 
in  both  cases.  When  d  <p  is  constant,  the  distance  between 
pairs  of  contiguous  ordinates  passing  through  the  -points 
A^a,J)}  &c.,  of  the  intersection  of  successive  adiabatics,  with 
any  isothermal,  A  B,  will  be  governed  by  the  law 


d_v  _  (dp'\  fdr_  \ 
dtf  ~  \~dr')  \dr 


140  IMPERFECT   FLUIDS.  [97«.] 

which  for  sensibly  perfect  gases  becomes,  observing  that, 
for  this  case,  r  =  T', 

d  v   _  2}>      r  _  P' . 
dv'        T'     p       J9' 

hence,  the  increments  of  tfte  volume  vary  inversely  as  the 
pressures  for  those  volumes,  or  directly  as  the  volumes. 

The  increments,  then,  become  indefinitely  large  as  the 
adiabatics  approach  indefinitely  near  the  axis  0  v  and  dimin- 
ish indefinitely  as  they  approach  the  axis  O  2). 

Let  these  adiabatics  be  nwribered  in  the  order  of  the 
natural  numbers,  beginning  with  any  arbitrary  number,  as 
7  for  A  <p^  then  will  a  e  be  8;  bd,  9,  &c.,  and  if  the 
terminal  one,  B  (?„  be  12,  the  number  of  spaces  between 
A  (p,  and  B  ^  will  be 

12  -  7  =  5, 
and  the  area 

ABDC=  5  X  AacC', 
similarly, 

CDFE=  5  xAacC; 

«fec.  &c. 

Generally,  if  tpt,  be  the  number  of  the  initial  adiabatic 
A  cpv  counting  from  any  arbitrary  zero,  <p*  of  B  <p^  then 

ABD  C=  (<pB-<pA}.Aac  C=  CDFE; 

and  if  tl  be  the  number  indicating  the  temperature  of-  the 
isothermal  A  JS,  and  rt  of  any  lower  isothermal,  as  E  F, 
then  will  the  area 

A  B  FE  =  (T,  —  T,)  (<p*  —  <pA).  AacC. 
Let  A  a  c  C  be  the  unit  of  measurement,  which  may  be 
arbitrarily  chosen,  and  we  finally  have 

A  B  FE  =  (T>  -  T,}  (q>B  -  <pA)  =  ,T,  X  A^B,          (132) 

in  which,  if  the  foot-pound  be  the  unit  of  heat,  the  ther- 
modynamic  function,  A<pB,  will  represent  the  NUMBER  of 
foot-pounds  of  heat  absorbed  by  the  substance  per  degree 


[97«.]  THE   THERMODYNAMIC    FUNCTION.  141 

of   temperature   while   expanding   isothermally    from   any 
adiabatic,  A  <p^  to  another,  B  cpv 
If  T^  =  o,  we  have 

<p,  A£<p,  =  TI  fa  -  9>A).  (132a) 

The  natural  zero-adiabatic  is  the  ordinate  Op,  but  be- 
tween that  and  the  initial  adiabatic  of  any  problem,  as  A  </>,, 
tiiere  will  be  an  infinite  number  of  adiabatics  including 
areas  equal  to  A  a  c  C  between  the  isothermals  B  A  and 
D  C',  hence  from  this  zero  the  number  between  A  and  B 
would  be  expressed  by  the  difference  between  two  infinites, 
thus, 

GO     —    00  , 

which  is  indeterminate.  An  arbitrary  zero-adiabatic  may 
be  assigned,  but  it  is  unnecessary  so  to  do,  since  it  is  only 
the  difference  of  the  thermodynamic  functions  that  is 
sought. 

The  form  of  the  expression  in  the  second  member  of 
equation  (132)  is  similar  to  that  expressing  the  area  of 
a  rectangle.  Thus,  suppose  that  in  measuring  a  rectangular 
field,  A  B  D  6Y,  a  point  is  arbitrarily  assumed  in  the  side 
A  B  prolonged,  from  which  the  corner  A  is  a  feet,  and  B,  1) 
feet ;  then  will  the  length  of  that  side  be  1)  —  a  feet  long  in- 
dependently of  the  position  of  the  point.  Similarly,  if  the 
corner  D  be  x  yards  from  a  point  in  the  line  of  B  D,  and 
B,  y  yards  from  the  same  point,  the  area  of  the  rectangle 
will  be 

A  B  DC  =  (b  —a)(x  —  y)  ft.-yds., 

the  unit  area  being  one  foot  wide  and  one  yard  long.  If 
differences -of  temperature  only  were  used  the  position  of 
the  absolute  zero  would  be  of  little  consequence. 

Equation  (132a)  furnishes  the  following  definition  :  The 
difference  between  the  numerical  values  of  the  thermodynam- 
ic  functions  corresponding  to  two  adiabatics  is  equal  to 
the  quotient  of  the  number  of  foot-pounds  of  heat  absorbed 


142  IMPERFECT   FLUIDS.  [97a.J 

or  rejected  in  passing  along  any  isothermal  from  one  of 
these  adiabatics  to  the  other,  divided  by  the  absolute  temper- 
ature corresponding  to  that  isothermal. 

If  the  path  of  the  fluid  is  not  an  isothermal,  the  same 
principle  is  applicable,  but  T  will  vary,  and  T  d  <p  cannot  be 
integrated,  but  the  difference  of  the  therinodynamic  func- 
tions may  be  determined  in  some  cases.  For  instance,  in 
Fig.  12,  we  have 


dH  „  dr 

•'•  —  —  =d<p=  Ay—  ; 

in  which,  if  K^  be  constant,  we  have 

<pB  —  9  A  =  /i"v  log  —  ; 

TA 

while  the  heat  absorbed  will  be 


When  the  path  is  arbitrary,  the  differential  expression  for 
the  thermodynamic  function  is 

dH  ^  dr         d 

--  =  d<p  =  Kv-r 

Clausius  calls  the  expression 
-dH 


/-a  11 

J     r 


"  the  entropy  of  the  body,"*  from  the  Greek  word  rponff, 
Transformation,  since  it  is  a  measure  of  the  rate  of  the 
transformation  of  heat  from  the  source  to  the  body  per  unit 
of  absolute  temperature  of  the  source,  in  passing  from  one 
adiabatic  to  another.  It  is  identical  with  Rankine's  thermo- 
dynamic function. 

It  will  be  observed  that  the  spaces  A  B  D  C,  O  D  F  E, 

&c.,  are  divided  into  the  same   number  of   strips  by   the 

adiabatics  through  A,  a,  J,  &c.;  hence,  cpB  —  (pA  has   the 

same  value  for  the  space  between  any  pair  of  isothermals, 

*  Mechanical  Theory  of  Heat,  p.  102. 


[98.]  LIQUID   AND   ITS   VAPOR   COMBINED.  143 

whether  equidistant  or  not.  If  the  state  a  approaches  A, 
ultimately  coinciding  with  it,  the  spaces  A  a  c  O,  C  c  e  E, 
&c.,  vanish  ;  in  other  words,  if  the  expansion  be  adiabatic> 
no  heat  will  be  absorbed  or  emitted,  and  for  this  case 

d  <f>  =  0; 
.  •  .  (p  =  constant  ; 

that  is,  the  entropy  —  or  the  thermodynamic  function  —  of 
an  adiabatic  is  constant,  and  this  is  the  characteristic 
equation  of  an  adiabatic.  For  this  reason  we  have  fre- 
quently used  the  symbol  q>  to  mark  the  adiabatic. 

This  property  may  be  put  in  contrast  with  a  property  of 
an  isothermal  in  the  following  manner  : 

That  property  of  a  substance  which  remains  constant 
throughout  such  changes  as  are  represented  by  an  isothermal 
line  is  the  temperature.  The  constant  property  is  that  of 
constant  heat. 

That  property  of  a  substance  which  remains  constant 
throughout  the  changes  represented  by  an  adiabatic  line  is 
the  Thermodynamic  function,  or  Entropy.  This  constant 
property  is  the  constant  rate  at  which  heat  must  be  ab- 
sorbed by  a  substance  per  unit  of  absolute  temperature  when 
the  path  of  the  fluid  is  from  any  point  on  an  adiabatic  to  a 
point  on  the  adjacent  one. 

98.  Liquid  and  its  vapor,  combined.  To  find 
the  differential  expression  for  the  heat  absorbed,  we  first 
find  the  heat  necessary  to  evaporate  the  d  x  part  of  one 
pound,  that  is,  a  weight  d  as.  For  d  v  volume  we  have 


and  if  the  volume  of  a  pound  of  saturated  vapor  be  v^  then 

d  v 

-  •  =  d  x. 
v* 

If  TFbe  the  pounds  of  fluid  of  which  d  TFbe  evaporated, 
then  if  d  II  be  the  heat  absorbed  for  an  increase  d  r  of 


144  IMPERFECT   FLUIDS.  - 

temperature  of  the  liquid  and  d  x  the  weight  evaporated, 

we  have 

r      dW*  7/e, 


or 

=  Cdr  +  dx  •  7/e,  (133) 

where  x  is  the  fractional  part  of  a  pound  of  the  substance 
vaporized.  Integrating  this,  observing  that  the  conditions 
of  the  problem  require  that  the  higher  temperature  rl  must 
be  that  at  which  the  quantity  x  is  evaporated,  we  have 

H=C(rl-T)  +  x.H.l.  (134) 

If  the  substance  be  water,  we  have 

C  =  J  =  778  foot-pounds, 
77e  =  1117880  -  544.6  r. 

Equation  (134)  is  sometimes  used  in  calorimeter  tests  for 
determining  the  amount  of  water  in  steam.  Thus,  to  find 
the  per  cent  of  water,  we  have  from  equation  (134) 


100(1-*)  =  100     .  -.-. 

(See  Addenda.) 

EXERCISES. 

1.  By  condensing  the  steam  from  a  boiler  into  a  reservoir 
of  water  it  was  found  that  600000  foot-pounds  of  heat  had 
been  imparted  to  one  pound  of  the  steam  above  the  temper- 
ature of  the  feed  water  ;  the  temperature  of  the  feed  water 
being  100°  F.  and  the  steam  from  the  boiler  320°  F.,  how 
much  liquid  water  did  the  steam  contain  ? 

Here  we  have 

r,  -  r,  =  220°, 
C(rl  -r,)  =  171160  ft.-lbs., 

H.  =  692000  "     "    nearly,  at  320°  F., 


{99.J  SPECIFIC   HEAT.  145 

which  is  the  foot-pounds  of  heat  that  the  steam  should  have  contained, 
above  100°  F.  if  it  had  all  been  evaporated,  but  the  test  showed  that 

//  =  600000  ; 
.  •  .  difference  —  263160  ; 

-.,-—  ,.-. 


2.  If  the  feed  water  be  100°  F.  and  the  temperature  of 
the  steam  be  338°  F.  and  the  heat  absorbed  above  that  of 
the  feed  water,  II  =  900000  foot-pounds,  required  the 
amount  of  water  suspended  in  the  steam. 

Here, 

C  (T,  -  T,)  =  238°  x  778 

=  185000 
He  =  683000  ft-lbs.  at  338°, 


sum  =  868000  "     " 

which  not  being  so  much  as  was  measured,  the  steam  must  have  been 
superheated. 

99.    The  specific  heat    of  saturated    vapor 

is  not  that  at  constant  pressure  nor  that  at  constant  volume, 
but  it  is  the  heat  necessary  to  raise  the  temperature  of  one 
pound  of  the  substance  one  degree  when  the  steam  remains 
continually  at  the  point  of  saturation.  Conceive  the  tem- 
perature of  the  entire  mass  to  be  increased  an  amount  d  r 
and  the  volume  an  amount  d  v  ;  then  will  the  heat  exist 
in  the  three  following  parts  : — 

1°.  The  heat  absorbed  by  the  liquid.      The  liquid  not 
evaporated  will  be  W  —  w,  using  the  notation  of  the  pre- 
ceding article,  and  the  heat  absorbed  by  it  will  be 
(W-w}Cdr. 

2°.  The  heat  absorbed  by  the  vapor.     Let  8  be  the  dy- 
namic specific  heat  of  the  saturated  vapor  of    constant 
weight,  then  will  the  heat  absorbed  by  it  be 
w  Sdr. 

3°.  An  additional  amount,  d  w,  of  the   liquid  will  be 
evaporated  both  on  account  of  the  enlargement,  d  v,  and  the 


146  IMPERFECT    FLUIDS.  [99.] 

increase  of  temperature,  d  T,  and  the  amount  of  heat  ab- 
sorbed will  be 

dw  -  J7e. 

Hence,  equation  (^1),  becomes 
W  •  dII=(W-'w}  Cdr  +  wSdr  +  dw  •  ne; 

.  • .  d  n  =  (1  -  x)  Od  r  +  x  S  d  r  +  JIe  d  x,  (135«) 
which  will  be  the  heat  absorbed  by  one  pound  of  the  entire 
substance  under  the  conditions  imposed.  All  the  quantities 
in  this  equation  have  been  determined  except  S. 

To  find  -6',  let  a  pound  of  liquid 
(water,  for  instance)  be  evapo- 
rated at  state  A  /  AD  being 
the  volume  when  the  o-th  part 
of  the  pound  is  evaporated,  neg- 
lecting the  volume  of  the  liquid ; 
A  F  the  volume  when  the  entire  pound  is  evaporated,  for 
which  x  =  1.  Let  state  B  be  d  r  higher  than  A,  B  m 
the  volume  when  the  sctli  part  is  vapor,  in  n  be  the  arbi- 
trary increment  of  increase  of  the  a*th  volume,  and  B  E 
the  volume  of  a  pound  of  the  vapor  for  the  pressure  O  B, 
then  d  x  —  m  n  -v-  B  E.  Draw  the  adiabatics  and  join 
n  D.  It  is  proposed  to  find  the  heat  absorbed  along  the 
path  n  D.  In  working  around  the  area  A  B  n  D  A  (or 
any  other  cycle)  the  heat  absorbed  minus  the  heat  emitted 
in  foot  pounds  will  equal  the  area  of  the  cycle. 

We  have 

A  D  -  xv, 

<Pt  A  B  q>t  =.  C  d  r  (heat  absorbed), 
<pt  D  A  q>i  —  x  //e  (emitted), 

<P9  B  m  <PI  =  x  7/e  -f-  sc  (—j — -  }  d  r  (absorbed), 
<?4  m  n  <pt  =  d  x  -  H,  -f 

last  term  being  a  difference  of  a  lower  order  than  the  pre- 
ceding term  will  be  neglected),  and  we  have 


[99.]  147 

9>4  in  n  q>6  =  d  x  •  IIe  (absorbed). 

g>3  D  n  cp6  =  d  II  (which  may  be  absorbed  or  emitted 
depending  upon  the  slope  of  n  D  /  we  will  consider  it  as 
emitted,  then  if  in  any  case  it  is  absorbed  the  algebraic  sign 
will  change). 

x  H 

— --  d  r  =  A  D  •  dp  =  x  v  •  dp  =  the  area  D  A  B  n    D 


.• .  dll=  CdT  +  x  (dH°  —  J*L)  d  T  -f-  //.  •  d  x  ;          (136) 
\  dr          r  J 

dr  \  dr  r   )  d  r 

which  is  the  specific  heat  of  a  fluid  in  which  the  (1  —  a?th) 
part  is  liquid  and  the  a?th  part  of  it  is  vapor,  the  path  being 
arbitrary. 

If  the  weight  of  vapor  remain  constant  during  the  change 
of  temperature,  then  d  x  =  0.  If  the  entire  pound  be  dry 
saturated  vapor  during  the  change  of  state,  then  a.  =  1  and 
d  no  =  0,  and  d  H  -r-  dr  will  be  the  specific  heat  of  the 
vapor  kept  at  the  point  of  saturation  throughout  the  change 
of  state ;  and  the  resulting  value  will  be  S  in  equation  (135«) ; 


.. 

d  r  dr 

or  in  heat  units 


substituting  Si  rom  (138)  in  (135a)  will  give  (136). 
For  water  c  —  1,  Ae  =  1436  -8  —  0  •  7  r  ; 

1436-8 
.•:.«>!-    -^-,  (HO) 

which  will  be  negative  for  all  values  of  r  less  than  1436°  F. 
above  absolute  zero,  or  976°  F.  above  the  zero  of  Fahren- 
heit's scale.  The  negative  value  may  be  thus  explained  :  — 
If  saturated  steam  be  expanded  in  a  non-conducting 
cylinder,  a  portion  of  it  will  condense,  giving  up  its 

For  an  analytical  solution  see  Sir  William  Thomson's  Math,  and  Phys.. 
Paper*,  Vol.  I.,  pp.  141-207  ;  Phil.  Mag.  (1852),  IV.;  Trans.  R.  Soc.  Ed., 
1851. 


148  IMPERFECT   FLUIDS.  [100.] 

heat  to  the  remainder  of  the  steam,  thus  maintaining  the 
temperature  corresponding  to  the  pressure  of  saturation  ;  and 
if  it  be  compressed  in  such  a  cylinder,  heat  must  be  abstracted 
if  the  pressure  and  temperature  continually  correspond  to 
those  of  saturation.  If  heat  be  not  abstracted  in  the  latter 
case  the  steam  will  be  superheated,  and  the  temperature  will 
exceed  that  corresponding  to  the  pressure  of  saturated  steam. 

In  regard  to  this  Rankine  said :  "  This  conclusion  (the 
liquefaction  of  steam)  was  arrived  at  contemporaneously  and 
independently  by  M.  Clausius  and  myself.  Its  accuracy 
was  subsequently  called  in  question,  chiefly  on  the  ground 
of  experiments  which  show  that  steam  after  being  wire- 
drawn, that  is  to  say,  by  being  allowed  to  escape  through  a 
narrow  orifice,  is  superheated,  or  at  a  higher  temperature 
than  that  of  liquefaction  at  the  reduced  pressure.  Soon  after- 
ward, however,  Professor  William  Thomson  proved  that 
these  experiments  are  not  relevant  against  the  conclusion  in 
question,  by  showing  the  difference  between  the  free  ex- 
pansion of  an  elastic  fluid,  in  which  all  the  power  due  to 
the  expansion  is  expended  in  agitating  the  particles  of  the 
fluid,  and  is  reconverted  into  heat,  and  the  expansion  of  the 
same  fluid  under  a  pressure  equal  to  its  own  elasticity, 
when  the  power  developed  is  all  communicated  to  external 
bodies,  such,  for  example,  as  the  piston  of  an  engine"  (Misc. 
Sc.  Papers,  p.  399). 

Professor  Clausius  said :  "  The  conclusion  that  the  spe- 
cific heat  of  saturated  steam  is  negative  was  drawn  by  Ran- 
kine and  by  myself  independently  at  about  the  same  time 
(Theory  of  Heat,  p.  135).* 

1OO.  Adiabatics  of  imperfect  gases.  This  con- 
dition requires  that  II  =  0,  .  • .  d  H  =  0  in  equations  (A\ 
giving 


*  Both  papers  were  read  in  February,  1850-Rankine's  in  Edinburgh, 
and  Clausius'  in  Berlin. 


1  100.]  ADIABATICS   OF   IMPERFECT   GASES.  149 


K 


In  order  to  integrate  the  first  of  these,  K^  and  ™  must 

be  known  functions  oi  r  and  v.  Jiv  not  only  depends  upon 
the  volume  but  is  not  a  known  function  of  r.  Even  grant- 
ing that  its  general  expression  is  given  by  equation  (105),  its 
determination  requires  a  knowledge  of  the  equation  of  the 
fluid,  and  that  can  be  known  only  empirically,  and  hence 
would  apply  only  for  the  range  of  the  experiments  upon 
which  the  formulas  were  based.  We  have,  however,  found 
for  carbonic  acid  gas,  and  for  all  other  fluids  investi- 
gated, that  the  specific  heat  at  constant  volume  for  a  con- 
stant state  of  aggregation  is,  without  a  large  error,  constant 
within  the  range  of  ordinary  experience  ;  and  similarly  for 
Kv  ;  hence,  representing  these  by  (7V  and  (7P,  respectively, 
we  have 


(Ml) 


in  which  y  must  be  constant  for  the  range  through  which  the 
specific  heats  are  considered  constant.  Assuming  equation 
(4)  as  the  general  equation  of  fluids,  and  considering  that 

&o  a,  a,  . 


dv\     dr 


J60  IMPERFECT   FLUIDS.  [100.; 

in  which  £>0,  &„  J,  are  constants  to  be  determined  by  experi- 
ment, we  have 

T?  v  =  R  t  —  — ° —  ~r &c. 

•*  V  T  V         t    V 

R   .      o,      .    2  &, 


and  (141)  becomes 


&  , 
y  d  v  =-  -TJ  aj), 

which  are  the  differential  equations  to  the  adiabatics  for 
imperfect  gases.  From  (143),  v  can  be  eliminated  by  means 
of  equation  (142),  resulting  in  an  equation  involving  r  and 
p  only ;  and  r  from  (143)s  by  means  of  the  same  equation  ; 
but  the  resulting  equations  will  be  too  complex  to  admit  of 
integration,  and  therefore  the  general  finite  equation  to  adia- 
batics is  unknown. 

It  is  customary  to  assume  that  the  equation  of  the  adia- 
batics for  such  superheated  vapors  as  are  used  for  engi- 
neering purposes,  as  steam,  is  of  the  same  form  as  that  for 
the  sensibly  perfect  gases,  at  least,  within  the  limits  used 
in  practice ;  and  hence  may  be  represented  by  the  equation 

p  v*  =  c,  (144) 

in  which  y  must  be  found  for  the  particular  substance, 
and  the  particular  state  of  that  substance. 

To  find  y  for  steam  considered  as  a  perfect  gas,  we 
found  in  Article  78  the  volume  of  a  pound  of  steam  at  212° 
JF.  under  the  pressure  of  one  atmosphere  to  be  26.5  cubic 


ADIABATICS    OF   IMPERFECT   GASES.  151 

feet  ;  hence,  if  it  followed  the  gaseous  law  down  to  32°,  the 
volume  at  the  latter  temperature  would  be 

v0  =  26.50  +  1.366  =  19.39  cu.  ft.; 
.-.  p0v0  =  19.39  X  2116.2  =  41033  ; 


A"p  =  0.48  X  778  =  373.44, 
/C  =  373.44  -  83.28  =  290.16, 

.-.y  =  f?  =1.3,  nearly;  (145) 

-Zly 

hence,  the  equation  of  the  adiabatic  for  steam,  considered  as 
a  sensibly  perfect  gas,  will  be 

p  v1-3  =  pl  v,  '•'  (146) 

This  value  of  y  is  used  for  superheated  steam  at  all  tem- 
peratures. 

But  steam  as  used  in  the  steam-engine  is  generally  more 
or  less  saturated,  for  which  case  Rankine  used  -a¥°-  for  the 
approximate  value  of  y^  so  that  for  such  cases  the  equation 
of  the  adiabatic  will  be 

pi^  —  p^^  (147) 

Rankine  was  the  first  writer  to  give  even  an  approximate 
equation  to  the  adiabatic  of  saturated  steam.  M.  G.  Schmidt, 
in  his  Theorie  des  Machines  d  Vapeur,  1861,  assumed  that 
steam  comported  like  a  perfect  gas,  and  so  assumed  y  —  1.4, 
a  value  entirely  without  foundation,  as  shown  by  equations 
(145)  and  (147),  and  which  that  author  later  abandoned. 

In  1863  Grashof  reviewed  the  question,  and  found  the 
mean  value  of  y  =  1.1354. 

Still  later,  Professor  Zeuner,  by  a  series  of  experiments 
in  which  the  initial  pressures  varied  from  1  to  4  atmospheres, 
final  pressures  from  £  to  2  atmospheres,  and  in  which  the 
specific  quantity  of  initial  vapor  (or  the  per  cent  of  the  fluid 


152  IMPERFECT   FLUIDS.  [100.] 

in  the  cylinder  that  was  vapor  before  cut-off)  was  0.70,  0.80, 
0.90 ;  found  results  from  which  he  concluded  that : 

The  value  of  y  is  dependent  chiefly  upon  the  initial 
specific  volume  of  the  vapor. 

That  it  is  nearly  constant  for  the  same  initial  state  of  the 
vapor  for  all  the  pressures  observed  from  one  to  four  at- 


That  the  value  of  y  may  he  represented  by  the  empirical 
formula 

y  =  1.035  +  0.100  a?,,  (148) 

in  which  a?,  is  the  initial  specific  quantity  of  the  vapor.  This 
formula  is  limited  to  values  of  a?,  between  0.7  and  1  (The- 
orie  Mecanique  do  la  Chaleur  (1869),  (329-335).  In  this 
formula,  if  xl  =  0.76,  that  is,  if  24  parts  in  100  of  the 
fluid  is  initially  water,  it  gives  y  =  1.111,  which  is  the  con- 
stant value  proposed  by  Rankine.  If  equation  (148)  can  be 
extended  to  values  of  a?,  much  less  than  0.7,  it  appears  that 
the  adiabatic  for  saturated  steam  approximates  more  and 
more  nearly  to  the  isothermal  of  the  perfect  gas  in  which 
y  =  1 ;  and  for  values  of  a?,  less  than  0.50,  the  two  curves 
will  nearly  coincide  within  the  ranges  of  expansions  used  in 
ordinary  practice.  Hence  the  curve  of  adiabatic  expansion 
of  wet  saturated  steam  approximates  to  that  of  the  equi- 
lateral hyperbola. 

But  when  we  consider  the  complex  nature  of  the  problem 
— the  temperature  of  the  surrounding  walls  being  modified 
by  the  nature  of  the  metal,  its  thickness,  its  exposure  exter- 
nally ;  the  time  of  the  exposure  internally  depending  upon 
the  piston  speed  ;  rendering  it  practically  impossible  to  realize 
exact  adiabatic  expansion — it  is  too  much  to  expect  any  em- 
pirical formula  to  cover  all  the  cases  of  approximate  adia- 
batic expansion  which  might  arise ;  and  we  conclude,  as  did 
Zeuner,  that  the  empirical  formula  of  Eankine,  equation 
(147),  is  sufficiently  exact  for  theoretical  or  practical  pur- 
poses when  the  initial  steam  contains  but  little  water. 


[100.] 


ADIABATICS   OF   IMPERFECT   GASES. 


153 


To  find  the  equation  to  the  adiabatic  for  saturated  vapor 
when  liquid  is  present,  make  d  H  =  0  in  equation  (136), 
and  it  reduces  to 


Integrating  between  initial  and  general 
limits  gives 

-.tf^-S-ii-fLS 


0        G 

FIG.    43. 


1  d  rl  d  r 

f  dp      i      ri  7        T\^?T 

.  •  .  x  v  =  (xlvl  -j±-±  +  C  log  -!- )  -: — 
V        ar,  f  /  dp 


(149) 


For  steam  C  becomes  J.     By  means  of  equation  (86),  if 
c  and  Ae  be  ordinary  thermal  units, 


t,          =         =  y 

" 


d  r        T 

and  (149)  may  be  written 


(U9«) 


In  this  solution  the  specific  volume  of  the  liquid  is  neg- 
lected, since  the  volume  of  both  the  liquM  and  the  vapor  is 
essentially  that  of  the  vapor.  In  equation  (149)  a?,  vl  = 
O  G,  Fig.  43,  will  be  the  volume  of  one  pound  of  the  satu- 
rated steam  and  water  at  the  beginning  of  expansion,  and 
a?  v  the  volume  of  the  steam  and  water  at  any  point  of  the 
expansion  B  C,  corresponding  to  the  temperature  T  or 

pressure  p.     Eliminating  ^  —  by  means  of  equation  (82), 
dp 

and  then  r  by  means  of  (81),  the  result  will  be  the  equation 
of  the  adiabatic  £  C\  but  the  second  member  will  be  too 


154  IMPERFECT   FLUIDS.  [101.] 

complex  to  be  of  practical  value  ;  and  the  approximate 
equation  of  Raiikine  (14:7)  will  be  used  instead.  An  im- 
portant theoretical  deduction  may  be  made  from  the  equa- 
tion in  its  present  form  ;  thus,  if  the  steam  be  dry  at  B,  the 
point  of  cut  off,  a?,  will  be  unity,  and  making  x  v  =  u,  we 
have  with  the  aid  of  (86), 

(150) 


which  is  positive  for  values  of  TV  less  than  1436°  F.,  the 
same  limit  that  makes  equation  (140)  negative.  This  shows 
that  the  volume  of  steam  and  water  will  be  less  than  the 
specific  volume,  v,  of  steam  only  at  the  temperature  T.  This 
is  due  to  condensation,  as  stated  in  Article  99. 

Let  the    initial   volume    of   steam  be  one  cubic  foot,  its 

weight  will  be  —  =  u\,  and  let  r  =  —  =  the  variable  ratio 

vi  v* 

of  expansion,  then  will  equations  (150)  and  (87)  reduce  to 

(152) 


by   means  of  which  the  ratio  of  expansion  may  be  com- 
puted. 

1O1.  Condensers.  A  condenser  consists  of  a  vessel 
kept  at  a  comparatively  low  temperature  by  means  of  cold 
water,  for  the  purpose  of  condensing  steam.  In  the  jet  con- 
denser a  liquid  spray  is  forced  into  the  vessel,  and  for  the 
surface  condenser  the  cold  liquid  circulates  about  the  vessel 
or  through  tubes  in  the  vessel,  producing  a  cold  surface. 
When  the  piston  of  the  engine  is  very  near  the  end  of 
its  stroke,  a  communication  being  made  between  the  steam 
end  of  the  cylinder  and  the  condenser  through  the  exhaust 
passage,  the  steam  rushes  into  the  condenser,  and  the  greater 


[101-1  CONDENSERS.  155 

part  of  it  is  suddenly  condensed  —  the  pressure  falling  to 
two  pounds  per  square  inch,  more  or  less.  Using  the  sub- 
script 2  to  indicate  the  conditions  at  the  end  of  the  stroke, 
and  3  for  those  in  the  condenser  at  the  end  of  the  operation, 
and  discarding  the  effect  of  molecular  changes  under  varying 
pressures,  thus  assuming  that  the  heat  abstracted  will  be  the 
difference  in  the  heats  in  the  initial  and  terminal  strokes 
(which  will  be  approximately  correct),  equation  (148)  will  gi  ve, 

H  =  J(T,  -  T3)  +  a?,  v,^l  -  x,v,^  ,          (153) 

t\  va 

for  water  and  for  the  Fahrenheit  scale,  and  is  the  heat 
abstracted  from  a  pound  of  steam  in  reducing  its  tempera- 
ture from  T7,  to  T3  degrees. 

The  steam  end  of  the  cylinder  will  remain  practically  at 
constant  volume  during  this  change,  and  neglecting,  as  be- 
fore, the  specific  volume  of  the  water  from  which  the 
steam  is  generated,  and  assuming  that  the  volume  of  the 
space  within  which  the  change  of  temperature  takes  place 
is  constant  during  the  change,  we  have 

a?,  v,  =  x3v3  =  ua  (154) 

and  the  preceding  equation  becomes 

H=J  (T,  -  TO  +  n,  &  -  ?*\  .  (155) 


In  a  continuously  working  engine  a  constant  mass  of 
vapor  remains  in  the  condenser  at  the  end  of  each  stroke, 
the  amount  condensed  being  equal  to  that  exhausted,  and 
H*  may  be  neglected  in  (155),  for  which  case  we  have 


Jf=J(Tz-  T3]  +  u,  _ 

The  pressure  of  the  vapor  in  the  condenser  determines 
its  temperature,  and  that  will  be  the  inferior  limit  of  tem- 
perature at  which  the  steam  will  be  worked. 


156  IMPERFECT   FLUIDS.  [101.] 


EXERCISE. 

Determine,  approximately,  the  amount  of  water  that  must 
pass  through  the  condenser  of  a  steam-engine  per  pound  of 
steam  exhausted,  having  given  Ty  =  300°  F.  the  temperature 
of  the  steam  in  the  cylinder  as  it  exhausts  into  the  condens- 
er, a-2  =  0.90  the  fractional  part  of  the  steai»  and  water  in 
the  cylinder  that  may  be  considered  as  pure  saturated  steam, 
the  pressure  in  the  condenser  two  pounds  per  square  inch 
absolute,  the  water  entering  the  condenser  at  60°  F.,  and 
leaving  it  at  100°  F. 

By  means  of  a  table  of  the  properties  of  saturated  steam,  Or  by  eqs. 

(78),  (85)  and  (95),  we  find,  using  approximations  to  the  larger  numbers. 

Temperature  of  the  condenser  for  2  Ibs.  per  sq.  in.,  T3  =  126 

From  the  problem,  7\  =  300 

Tt  -  Tt  =  174  ; 
.  '  .  J(T*  -  T3)  =  174  X  778  =  135400  ft.-lbs. 

Total  heat  of  the  steam  at  300°  will  be  778  X  1173  =  912600 

Heat  in  the  water  above  32°,  778  X    270  =  210000 


Difference,  Hf,  =  702600. 

The  table  gives,  for  the  specific  volume  of  the  steam  at  300°, 
v.,  =  6.2  cu.  ft. ; 

...f  =  113000. 

Total  heat  of  steam  at  126%  778  X  1120  =  871400 

Heat  in  the  water  above  32°,  778  X  94      =    73100 


Difference,  Htt  =  798300 
The  tabular  value  for  vt  is 

vs  =  172    cu.  ft.; 


These  values  in  equation  (155)  give 

H=  135400  -f  604600  =  740000  ft.-lbs. 

The  water  supplied  to  the  condenser  being  raised  through  100  —  60  =  40 
degrees,  the  quantity  required  will  be 


[102.]  ISODIABATIC   LINES.  157 

740000 

9  =  77g  x  4Q  =  24  pounds,  nearly. 

Equation  (155a)  gives  24.7  Ibs.  ;  that  is,  a  condensing  engine  running 
with  stearn  at  52  pounds  gauge  pressure  will  require  about  25  pounds 
of  water  for  the  condenser  for  every  pound  of  steam  condensed  if  the 
temperature  of  the  water  be  raised  40  degrees.  If  a  greater  difference 
of  temperature  of  the  water  at  arriving  and  leaving  be  allowed,  it  would 
require  less  water,  or  if  the  gauge  pressure  be  higher,  it  will  require  more 
water  for  the  same  difference  of  temperature. 

The  numerical  computation  of  (155)  will  be  facilitated  by 
a  table  of  the  latent  heat  of  evaporation  per  cubic  foot,  since 


1O2.  Isodiabatic  Lines.  Let  CN  and  B  M,  Fig.  44, 
be  any  two  isothermals  cut  by  an  arbitrary  path  A  D.  In  pass- 
ing from  A  to  D  a  certain  amount  of  heat  will  be  absorbed, 
represented  by  the  area  between  D  A  and  two  adiabatics 
drawn  from  A  and  D  respectively,  as  shown  in  Article  34. 
It  is  possible  to  find  another  path,  C  B,  in  working  along 
which  the  same  amount  of  heat  will  be  emitted  as  was  ab- 
sorbed along  A  D.  To  prove  this,  conceive  an  indefinite 
number  of  isothermals  between  C  N  and  B  M,  and  at  the 
points  of  division  with  A  D  draw  adiabatics  ;  then  find 
a  point  near  C,  which  call  z,  on  the  isothermal  next  below 
C  J),  such  that  when  joined  with  C  the  area  included  be- 
tween z  C  and  two  adiabatics  through  z 
and  C\  respectively,  will  equal  that  be- 
tween the  corresponding  pair  at  D. 
Proceed  in  this  manner  with  the  next 
isothermal,  and  so  on  to  B\  then  will 
the  area  between  B  C  and  the  adiabatics 
through  B  and  C  respectively  equal  the 
area  between  A  D  and  the  adiabatics 
through  A  and  D  respectively,  which  was  to  be  proved.  The 
lines  D  A  and  B  C  are  called  isodiabatics  in  reference  to 
each  other  (Rankine's  Misc.  Sc.  Papers,  p.  345  ;  Steam- 
Engine,  p.  345). 


158  IMPKUFECT   FLUIDS.  [102.] 

To  find  the  analytical  condition,  conceive  C  JV  and  B  M 
to  be  consecutive  isothermals,  then  the  heat  absorbed  in 
passing  from  A  to  D  will  be,  from  equation  (A\ 

d  11=  Crdr 

and  that  emitted  along  C  B, 

d  11=  C'vdr  + 

which,  according  to  the  conditions  of  the  problem,  are  to 
be  equal,  giving 


(£)<•=  ( 


which  relation  is  independent  of  the  specific  heat  of  the  sub- 
stance. For  sensibly  perfect  gases  we  have 

fdp\  _  p  _  R 

\d^l     T     T 

(dp\  _P>  _  B 
(dr'J  ~^~^' 

and  by  substituting  above  and  integrating,  we  have 

v  =  B  v 
or 

—  =  A.  a  constant ;  (157) 

P, 

that  is,  the  ratio  of  the  pressures,  or  of  the  volumes,  at  tlie 
respective  points  where  the  successive  isothermals  cut  the 
curves  A  D  and  B  C  must  le  constant. 


CHAPTER  IY. 

HEAT    ENGINES GENERAL    PRINCIPLES. 

103.  Efficiency. — Heat  engines,  in  practice,  work  in 
cycles,  and  when  running  under  uniform  conditions,  the  suc- 
cessive cycles  will  be  identical,  in  which  case  the  total  effect 
will  be  that  produced  in  one  cycle  multiplied  by  the  num- 
ber of  cycles.     It  is,  therefore,  important  to  investigate  the 
properties  of  one  cycle. 

The  efficiency  of  a  plant  is  the  ratio  of  the  work  which 
the  plant  can  produce  to  that  of  the  energy  supplied.  Thus, 
if  the  plant  consist  of  a  furnace  and  engine,  it  is  the  ratio 
of  the  work  it  can  do  to  the  theoretical  energy  of  the  fuel 
supplied  to  the  furnace. 

The  efficiency  of  an  engine  is  the  ratio  of  the  work  it 
can  do  to  the  energy  of  the  heat  absorbed. 

In  case  of  an  hydraulic  machine,  it  is  the  ratio  of  the  work 
it  can  do  to  the  theoretical  energy  of  the  waterfall. 

The  measure  of  the  efficiency  does  not  involve  the  mag- 
nitude of  the  machine,  and,  hence,  is  only  an  incidental 
element  in  proportioning  the  engine.  If  one  pound  of  air 
when  worked  in  a  cycle  will  produce  a  given  amount  of 
work,  two  pounds  will  produce  twice  as  much  when  worked 
in  a  similar  cycle.  The  proportions  of  an  engine  having  a 
given  efficiency  depend  upon  the  amount  of  work  to  be 
done  in  one  cycle. 

104.  Perfect  elementary  heat    engine. — An 
engine  receiving  all  its  heat  at  one  temperature  and  rejecting 
heat  at  one  lower  temperature,  must  pass  through  its  series 


160 


HEAT   ENGINES. 


[104.] 


M, 


of  changes  of  pressure  and  volume  according  to  Carnot's 
cycle.     Such  an  engine  is  reversible.     No  such  engine  can 
be  constructed  or  operated,  but  as  it  would  give  the  high- 
est theoretical  efficiency  of  any  engine 
working  between  the  temperatures   of 
the  source  and  refrigerator,  it  serves  as 
a  theoretical    standard  of   comparison, 
and  is  referred   to  as  a   Perfect   Ele- 
•M»    mentary  Heat  Engine. 

Let  AI  A,  Bt  J?,,  Fig.  45,  represent 
a  Carnot's  cycle,  according  to  which  the 
engine  receives  all  its  heat  at  the 
temperature  T,,  being  the  temperature  of  the  isothermal 
A,  A^ ;  and  rejects  heat  only  at  the  temperature  ra,  being 
the  temperature  of  the  isothermal  B,  Bv  Then  will  the 
heat  absorbed  in  expanding  from  state  A,  to  A,  at  the  con- 
stant temperature  rl  be,  according  to  equation  (-4),,  page  48, 
since  d  r  will  be  zero, 


H,  =  0  -f  rl 


and  the  heat  absorbed  along  the  adiabatic  At  B^  will  be 


v     v  v4 
FIG.  45. 


and  the  heat  rejected  along  the  isothermal  B^  B^ 


and  along  the  adiabatic  B^  At, 


=    -    r/rv,Zr-    r\(*P\dv; 
JT,  Jv,       ^l*' 


and  the  sum  of  these  will  give  the  heat  transmuted  into  ex- 
ternal work,  since  the  cycle  is  complete  ;  hence, 


[104.]         PERFECT   ELEMENTARY   HEAT   ENGINE.  161 

(158) 


The  efficiency,  according  to  the  preceding  article,  will  be 
E-  H*  ~  H*  -r*-r*-        T*~T*  (159) 

"HT     ~^~  -j\~+m^' 

Since  equations  (A)  are  general,  and  applicable  to  all 
substances,  the  result  must  be  equally  general  ;  hence,  the 
efficiency  of  the  perfect  elementary  engine  depends  only 
upon  the  highest  and  lowest  temperatures  between  which, 
it  is  worked,  and  is  independent  of  the  nature  of  the 
working  substance. 

If  iron,  or  any  other  solid,  could  be  worked  between  the 
temperatures  r,  and  r2,  according  to  Carnot's  cycle,  it  would 
be  just  as  efficient  as  if  the  substance  were  the  most  perfect 
gas.  The  range  of  volumes  through  which  solids  expand 
and  contract  is  small,  so  that  the  work  done  in  a  cycle 
would  be  comparatively  small,  and  the  changes  of  temper- 
ature are  so  slow  as  to  preclude  the  use  of  such  substances 
in  the  construction  of  heat  engines.  But  this  fact  does  not 
affect  the  efficiency  of  the  cycle. 

The  highest  temperature  at  which  the  engine  works  can- 
not exceed  that  of  the  source,  for  it  is  an  axiom  that  heat 
cannot  of  itself  flow  from  a  hot  body  to  one  still  hotter, 
a  principle  stated  by  Clausius  (Theory  of  Heat,  p.  78). 

Neither  can  it  be  worked  at  a  lower  temperature  than 
that  of  the  refrigerator,  for  it  is  held  as  an  axiom  that  a 
heat  engine  cannot  be  worked  at  a  lower  temperature  than 
that  of  the  coldest  of  surrounding  bodies,  a  principle  stated 
by  Thomson  (Math,  and  Phys.  Papers,  p.  181).  These 
axioms  are  the  same  in  substance,  and  originally  were 
stated  independently  by  the  respective  authors. 

If  any  of  the  heat  absorbed  is  at  a  lower  temperature 
than  r1?  while  all  is  rejected  at  r»  the  efficiency  will  be  less 


162 


HEAT  .ENGINES. 


[104.] 


than  if  it  were  all  absorbed  at  the  higher  temperature.  To 
show  this,  let  T3  be  the  constant  temperature  of  the  second 
source,  then  we  would  have 


=  *> 

^ 


and  the  efficiency  would  be 


fft  +  IF, 

which  is  less  than  the  value  of  equation  (159)  so  long  as 
H\\e>  less  than  fft.  A  reversible  engine  has  the  highest 
efficiency  for  the  heat  utilized,  and  the 
perfect  elementary  heat  engine  has  the 
highest  efficiency  of  any  engine  work- 
ing between  the  same  limits  of  temper- 
ature. 

The  principle  of  efficiency  is  applied 
in  the  same  manner,  whatever  be  the 
FIG  46 path  of  the  fluid.    Thus,  if  the  cycle  be 
A  a  B  d  A,  Fig.  46,  A  M  and  B  N 
adiabatics  indefinitely  extended,  then,  according  to  Article 
34,  we  have 
H,  =  MAaBN, 
II,  =  MAdBN-, 

.  v_  MAaBN-  MAdBN     AaBdA  __  U,- 

f.Aa. 


MAaBN 


, 


If  the  indicator  card  of  the  steam-engine  were  A  B  C  J, 
Fig.  47,  in  which  A  B  is  the  steam  line  of  constant  tem- 


[104.]        PERFECT   ELEMENTARY   HEAT   ENGINE.  163 

perature,  r^  B  C  the  expansion  line  of  no  transmission  of 
heat  extended  until  the  pressure  falls  to  that  of  the  back 
pressure,  C  J  the  back  pressure  line 
of  constant  temperature,  T^  and  J  A  the 
compression  line  of  no  transmission  of 
heat  being  made  to  pass  through  the  ini- 
tial state,  A,  then  will  the  efficiency  be 
r.-r,  _  Tt  -  T, 

rt 

as  before. 

In  Fig.  45  a  constant  quantity  of  air  is  supposed  to  remain 
in  the  cylinder  of  the  engine  during  the  changes  forming  the 
cycle,  but  in  the  steam-engine  the  heat  is  carried  into  the  cylin- 
der with  the  steam,  so  that  the  mass  of  steam  increases  with 
the  stroke  from  A  to  JB,  Fig.  47 ;  from  I>  to  C  the  mass  re- 
mains constant ;  at  Cfhe  exhaust  is  open,  communicating  with 
the  refrigerator,  and  remains  open  until  the  piston  reaches  «/, 
at  which  point  the  exhaust  is  closed,  and  the  mass  of  steam 
remaining  in  the  cylinder  at  J  remains  constant  throughout 
the  compression  J  A.  At  the  completion  of  the  cycle  the  fluid 
in  the  cylinder  at  the  state  A  will  have  the  initial  pressure 
and  volume,  but  since  the  changes  of  state  are  not  effected 
with  a  constant  mass  of  fluid  the  operation  will  not  be  that 
of  a  Carnot's  cycle,  and  the  above  expression  for  efficiency 
will  not  be  applicable. 

The  only  theoretical  mode  of  improving  the  efficiency 
of  the  elementary  engine  is  to  increase  the  range  of  tem- 
peratures between  which  it  is  worked. 

It  does  not  follow  from  this  principle  that  different 
substances  worked  between  the  same  limits  of  pressure 
will  be  equally  efficient,  for  pressures  are  not  proportional 
to  the  absolute  temperatures,  except  for  the  sensibly  per- 
fect gases.  If,  however,  the  operation  be  in  a  Carnot's 
cycle,  the  temperatures  corresponding  to  the  pressures 
being  found,  equation  (159)  will  be  applicable. 


164  HEAT   ENGINES.  [104.] 

It  might  be  urged  that  some  work  would  be  expended 
in  forcing  the  mass  of  steam  into  and  out  of  the  cylinder, 
thereby  producing  less  external  work  than  the  same  heat 
would  do  in  case  the  changes  were  produced  with  a  con- 
stant mass  of  fluid  in  the  engine.  In  regard  to  this  point, 
it  is  sufficient  to  observe  that,  if  the  argument  be  valid, 
the  energy  so  absorbed  is  too  insignificant  compared  with 
the  heat  energy  of  the  fluid,  to  be  considered. 

Actual  engines  do  not  produce  the  indicator  diagrams 
here  assumed,  and,  hence,  must  be  made  the  subject  of 
special  investigation. 

EXERCISES. 

1.  In  an  ideal  elementary  engine  working  one  pound  of 
air,  if  the  lowest  pressure  be  that  of  one  atmosphere, 
2116.2  Ibs.  per  square  foot  at  Bn  Fig.  45,  the  absolute 
temperature  of  the  refrigerator  ra  =  550°  (T,  =  89.34°  F.), 
that  of  the  source  r,  =  950°  (T,  =  489.34°  F.),  and  the 
volume  swept  through  by  the  piston  during  each  single 
stroke  12  cubic  feet;  find  the  greatest  and  least  volumes  of 
the  air  in  the  cylinder,  the  power  developed  in  one  end  of 
the  cylinder  during  one  cycle — or  double  stroke  of  the 
piston — the  heat  absorbed,  and  the  efficiency. 

To  find  the  largest  volume,  vt,  we  have,  equation  (3), 
v   _  53-21  T*  _  53.21  X  550  _  13  83  cu  ft 
p,  2116.2 

To  find  p,  and  va  the  adiabatic  A,  B»  equation  (41), 
gives 

7^1  3.463 

=  2116.2  (^  }         =  14045  Ibs. 


2.463 
=  13.83  =  3.60  cu.  ft. 


[104.J         PERFECT   ELEMENTARY   HEAT   ENGINE.  165 

To  find  the  least  volume,  #15  the  problem  gives 
v4  —  vl  =  12  ; 

.  •  .  Vl  =  13.83  —  12  =  1.83  cu.  ft. 
And  the  isothermal  A^  A3  gives,  equation  (3), 

p,  v,  =  53.21  X  950  =  50550  ft.-lbs. 
.  •  .  pl  =  27630  Ibs.  per  sq.  ft. 

=  191.9  Ibs.  per  sq.  in. 
Similarly, 

£  =  ?!*  =£  =  ?!?  =  1.97; 

1\       v3      p,       v, 

.'.p,=  41  62  Ibs. 

v2  =  7.03  cu.  ft. 
The  heat  absorbed  will  be,  equation  (36), 

pi  Vl  log  -*  =  34207  ft.-lbs. 
The  heat  rejected  will  be 

^  X  34207  =  19804  ft.-lbs.  ; 

Lu 

and,  hence,  the  work  done  in  one  cycle  will  be 
34208  -  19804  =  14404  foot-pounds, 

independent  of  the  time. 
The  efficiency  will  be 

14404 

34208  =  °'42' 

according  to  which  more  than  half  the  energy  of  the  heat 
is  rejected  by  the  engine.  The  ratio  of  the  greatest  to 
the  least  volumes  is 


and  of  pressures, 


-4  =  Ti,  nearly, 


=  13. 

1\ 


166  HEAT     EXGIXES.  [105.] 

2.  In  the  preceding  exercise,  what  must  be  the  area  of 
the  piston  in  order  to  operate  one  pound  of  air  between 
the  limits  assigned,  the  stroke  of  the  piston  being  six 
feet. 

'3.  In  Exercise  1,  if  the  engine  make  20  revolutions  per 
minute,  what  will  be  the  horse-power  developed  on  one  side 
of  the  piston-? 

4.  If,  in  Exercise  1,  two  pounds  of  air  had  been  used, 
and  the  lowest  pressure  that  of  one  atmosphere,  the  tem- 
peratures being  the  same  as  those  given  in  the  exercise, 
what  would  have  been  the  greatest  and  least  volumes  of  air, 
the  volume  swept  through  by  the  piston  being  24  cu.  ft.? 
Would  the  efficiency  be  the  same  ?     Would  the  work  have 
been  the  same  for  the  same  volume  swept  through  by  the 
piston  ? 

5.  If  in  an  elementally  air  engine  the  highest  pressure  be 
150  pounds  per  square  inch,  the  highest  temperature  450°  F., 
the  lowest  pressure  14.7  pounds  per  square  inch,  and  lowest 
temperature  60°  F.,  what  will  be  the  volume  swept  through 
by  the  piston  per  pound  per  stroke  ? 

1O5.  Regenerators  consist  of  a  chamber  well  filled 
with  thin  plates  of  metal  so  arranged  as  to  present  a  large 
surface  to  the  fluid  and  offer  as  little  resistance  to  its 
passage  as  possible.  The  fluid,  after  escaping  from  the 
engine  by  passing  through  this  chamber  to  the  refrigerator, 
gives  up  a  portion  of  its  heat  to  the  metal  plates,  the  re- 
frigerator finally  absorbing  the  heat  which  is  permanently 
rejected  ;  after  which,  by  passing  back  through  the  chamber 
and  being  at  a  lower  temperature  than  during  its  former 
passage,  it  absorbs  heat  from  the  plates,  thus  requiring  a  less 
amount  from  the  source  in  order  to  raise  it  to  the  required 
temperature.  During  the  flow  of  the  air  from  the  cylinder 
the  plates  act  as  a  refrigerator,  by  abstracting  heat  from  the 
gas ;  but  during  the  return  of  the  gas  they  act  in  the  oppo 
site  sense,  and  hence  become  regenerators. 


[105.]  REGENERATORS.  167 

If  the  temperature  changed  by  insensible  degrees  in  the 
regenerator,  the  efficiency  would  be  unaffected,  but  such  not 
being  the  case,  they  cause  a  loss  of  5  or  10  per  cent,  even 
when  well  proportioned.  Their  great  advantage  consists  in 
reducing  the  size  of  the  cylinder,  as  will  appear  in  the  fol- 
lowing exercise. 

Assume  that  heat  is  absorbed  at  one  temperature  and  re- 
jected at  another,  as  in  the  preceding   case,  but    that  the 
change  of  pressure  from   one  isothermal 
to  the  other  is  effected  at  constant  volume 
by  passing  the  air  through  a  regenerator, 
in  which  case  the  indicator  diagram  will 
be  represented  by  Fig.  48.     If  rl  be  the 
temperature  of  the  isothermal  B  C,  r^  of 
A  _£>,  Cv  the  specific  heat  at  constant  vol- 
ume, the  heat  absorbed  in  passing  from  A 
to  B  to  C,  if  the  operation  were  reversible,  would  be,  equa- 
tion (5),,  page  50, 

II,  ^  C\  (r,  -  r.)  +  ^  ^ 

heat  rejected, 


.-.//,-//,  =  B  (r,  -  r,) 


which  is  the  mechanical  energy  expended.  But  in  deter- 
mining the  efficiency,  the  loss  of  heat  in  passing  to  and 
fro  through  the  regenerator  must  be  added  to  U1  ;  since 
that  amount  of  heat  must  be  drawn  from  the  source  and 
is  not  accounted  for  in  the  preceding  value  of  H^  and,  rep- 
resenting  this  by  the  expression 

»#v  ft    -    *,}, 


168  HEAT  ENGINES.  [104.] 

in  which  n  is  a  fraction,  <7V  =  0.169  X  778  =  131,  we  have 
for 

U 
the  efficiency  =  ^  +  13J  n  ^  _  ^ 

in  which  n  will  be  -fa,  or  -^,  or  whatever  fraction  repre- 
sents the  lieat  lost  by  the  regenerator. 


EXERCISE. 

In  an  air  engine  with  a  regenerator  producing  changes  at 
constant  volume,  let  pt.  vt ;  p»  v, ;  r,,  r,,  be  the  same  as  in 
the  first  of  the  preceding  exercises  ;  determine  the  ratio  of 
the  pressures  and  volumes. 

Considering  the  engine  as  perfect,  the  work  done  will 
be  the  same  as  in  Exercise  1,  page  164,  for  the  expansion 
during  the  absorption  of  heat  must  be  the  same. 

We  will  have, 

p,  =  27630,     j>,  =  14045 ; 
V|  =  1.83  =  vn  v,  =  3.6.  =  va  Fig.  48 ; 
then 

p<  -.-=  p,  L*  =:  li  x  14045  =  8132  Ibs., 

ps=pl-  =  —  X  27630  =  15996  Ibs.; 
ri   iy 

.-.£«=  1.97, 

vi 

£  =  3.40. 
P. 

Comparing  these  results  with  the  exercise  referred  to, 
it  appears  that  the  greatest  volume  in  that  case  was  nearly 
4  times  the  greatest  volume  in  this ;  hence,  the  volume  of 
the  cylinder  with  the  regenerator,  under  the  conditions 
imposed,  need  be  only  about  one-fourth  as  large  as  without 


[106-8.]  STEAM-ENGINE.  169 

106.  Air  engines   have   been   made    in   which   the 
changes  of  temperature  have  been  effected  at  nearly  con- 
stant pressure,  and  others  in  which  the  change  takes  place 
at  nearly  constant  volume.     These  conditions  require  special 
forms  of  mechanism,  which  will  be  considered  further  on  •; 
but  the  work  performed  in  a  cycle  may  be  computed  from 
the  indicator  card,  as  in  Articles  104  and  105. 

107.  Heat  engines,  whether  of  air,   or  steam,  or 
other  vapor,  are  assumed  to  transform  a  certain  amount  of 
heat  energy  into  work  independent  of  the  mechanism  in- 
volved.    That  is,  aside  from   the   friction  of    the   engine 
wastes  due  to  leaks  arid  clearance,  it  is  immaterial  whether 
the  engine  be  single-acting,  double-acting,  reciprocating,  os- 
cillating, rotary,  disk,  trunk,  compound,  or  any  other  of  the 
many  forms  of  engines  used  ;  the  work  done  will  be  the  same 
in  all  the  engines  by  the  same  fluid  worked  between  the 
same  limits  of  temperature. 

Therefore,  considering  the  engine  as  a  Jceat  engine  only, 
we  have  only  to  consider  the  thermal  changes  produced  in 
the  working  fluid  during  a  complete  cycle,  involving  the  tem- 
perature of  the  feed  water,  and  the  initial  and  final  tem- 
peratures in  the  cylinder.  But  &.s,apiece  of  mechanism,  the 
several  forms  have  their  mechanical  advantages,  which  must 
be  considered  in  the  light  of  practical  mechanism.  All  the 
details  of  the  engine,  such  as  the  valve  mechanism,  the  size 
of  the  bearings,  the  strength  of  the  parts,  compactness,  etc., 
belong  to  constructive  mechanism,  and  are  treated  of  in 
works  which  consider  these  engines  as  machines. 

In  order  to  analyze  a  heat  engine  it  is  necessary  to  know 
the  law  according  to  which  it  receives  and  rejects  heat ;  and 
since,  in  actual  engines,  all  these  laws  are  not  known,  as- 
sumptions in  regard  to  them  are  made  which  are  supposed 
to  be  approximately  correct. 

108.  Steam-engine. — Steam  in  the  cylinder  works 
under  such  a  variety  of  conditions  that  a  complete  analysis 


170  HEAT    ENGINES.  [108.] 

requires  the  consideration  of  several  hypotheses.  Thus, 
steam  may  be  superheated,  in  which  case  it  will  expand, 
approximately,  like  a  perfect  gas ;  or  it  may  be  saturated, 
in  which  case,  by  expanding  without  transmission  of  heat,  it 
may  remain  constantly  at  the  point  of  saturation  ;  or  by  means 
of  a  steam  jacket,  the  steam,  by  being  constantly  supplied 
with  heat,  may  be  considered  as  dry  saturated  steam.  The 
curve  of  expansion  may  be  too  complex  to  be  analyzed 
with  great  exactness.  "When  steam  enters  the  cylinder  it  may, 
.  and  generally  will,  be  hotter  than  the  walls  of  the  cylinder, 
and  give  up  heat  to  the  walls,  thus  reducing  the  pressure, 
even  if  it  does  not  actually  condense  any  of  the  steam  ;  and 
as  the  steam  becomes  cooler  by  expansion,  the  walls  of  the 
cylinder  will  give  up  heat  to  the  steam,  thus  raising  its 
pressure  at  the  latter  part  of  the  stroke.  The  water  in  the 
cylinder,  if  any,  may  also  be  re-evaporated.  In  either  case 
the  restored  heat  taking  place  near  the  end  of  the  stroke 
does  not  compensate  for  the  loss  at  the  beginning,  for  the 
former  can  act  through  only  a  small  part  of  the  stroke,  and 
as  soon  as  the  exhaust  opens  the  restored  heat  escapes  with 
the  stea/n  and  is  lost.  Water  in  the  cylinder  may  result 
from  condensation  of  the  saturated  steam,  as  shown  in 
Article  99,  or  it  may  be  carried  over  from  the  boiler  with 
the  steam  in  the  form  of  very  small  drops,  as  a  spray.  If 
the  cylinder  be  jacketed  the  walls  will  be  kept  at  a  more 
nearly  uniform  temperature,  and  thus  condensation  in  the 
cylinder  be  prevented,  which  is  a  great  gain  in  the  working 
of  the  engine.  Condensation  in  the  steam  jacket  does  not 
affect  the  working  of  the  engine.  The  refinements  result- 
ing from  these  numerous  conditions  are  beyond  the  reach  of 
analysis,  because  the  laws  governing  their  actions  are  un- 
known. This  fact,  however,  is  not  seriously  prejudicial  to 
analysis,  for  the  hypotheses  assumed  agree  so  nearly  with 
actual  cases  as  to  give  results,  not  only  approximately  cor- 
rect, but  so  nearly  correct  as  to  be  reliable  in  ordinary 


[109.]  IDEAL   STEAM   DIAGRAM.  171 

practice.  If,  however,  it  becomes  necessary  to  investigate 
these  refinements,  or  so-called  exceptional  conditions,  the 
problem  of  the  steam-engine  in  this  respect  ceases  to  be 
analytical,  and  is  essentially  empirical.  It  must  not  be  in- 
ferred that  theory,  even  in  this  case,  is  useless,  or  is  to  be 
ignored,  for  it  is  only  by  theory  that  exceptions  are  known. 
Theory  gives  the  first  grand  approximation  to  the  truth, 
when,  by  comparing  the  results  with  actual  cases,  the  de- 
fects in  the  theory  become  known,  and  thus,  in  turn,  furnish 
the  means  of  correcting  or  amending  the  original  theory  ; 
after  which  a  second  and  nearer  approximation  may  be  made, 
and  so  on,  bringing  the  results  of  theory  and  of  practice  more 
nearly  to  an  agreement.  A  consideration  of  these  many 
conditions  demands  a  special  treatise  ;  we  will  consider  only 
a  few  special  cases. 

1O9.  Ideal  steam  diagram.— Let  A  B  CEF  be 
an  ideal  diagram  of  a  steam-engine,  A  B 
being  the  steam  line  at  constant  tem- 
perature and  pressure,  BC  the  expan- 
sion line,  C  E  the  fall  in  pressure  at  the 
end  of  the  stroke,  due  to  the  sudden 
opening  of  the  exhaust  passage,  E  T^the  * ' ' ' 


C 


back  pressure  line,  0  H  the  line  of  ab-      O       0- 

solute  zero  of  pressures  ;  then  O  A  =  GB  FIG-  49. 

will  be  the  total  forward  pressure  up 

to  the  point  of  cut-off,   C  H  the  forward   pressure  at  the 

end  of  the  stroke,  II  E  =  0  J^the  back  pressure. 

The  admission  line  A  B  is  an  isothermal  of  constant 
pressure,  and  in  this  respect  resembles  the  case  described 
in  Articles  74  and  77,  in  which  a  liquid  was  evaporated 
under  constant  pressure  at  a  constant  temperature.  In  that 
case  more  and  more  liquid  was  evaporated,  producing  more 
and  more  steam,  as  the  volume  increased,  while  here  more 
and  more  steam  enters  from  the  source  as  the  volume  in- 
creases. We  might  proceed,  as  with  the  perfect  engine,  to 


172  HEAT   ENGINES.  [HO.j 

find  the  heat  absorbed  and  rejected  throughout  the  cycle, 
and  take  the  sum  ;  but  it  is  customary  to  tind  the  results 
directly  in  terms  of  pressure  and  volume. 

The  ideal  diagram  is  one  freed  from  all  irregular  and  dis- 
turbing causes,  such  as  late  opening  for  admission,  initial 
expansion,  wire  drawing  at  the  point  of  cut-off,  slow  closing 
of  the  port,  irregularities  in  the  expansion  line  B  C,  too 
early  opening  of  the  exhaust  near  C,  a  want  of  sufficient 
opening  at  E,  and  of  compression  near  F ';  but  such  a  dia- 
gram represents  the  greater  part  of  the  work  done,  and  by 
applying  theory  to  it  a  result  approximately  correct  will  be 
obtained. 

1 1 0.  Isothermal  expansion. — Assume  that  the 
steam  is  superheated  and  the  cylinder  steam  jacketed,  then 
will  the  expansion  line  be  nearly  isothermal.  Assume  it  to 
be  exactly  so,  and  let 

pl  =  0  A  —  G  B,  Fig.  49,  be  the  initial  pressure, 
j).t  —  II  C,  the  terminal  pressure, 
pt  =  II E,  the  back  pressure, 
p  =  any  ordinate  to  B  C, 

«,  =  0  G  =  the  volume  occupied  by  one  pound  of 
steam  in  the  cylinder  up  to  the  point  of  cut-off, 
vt  =  O  H,  the  volume  of  one  pound  at  full  stroke, 
r   =  v,  -f-  vl  =  ratio  of  expansion, 
i)    =  any  volume  between  G  and  H, 
pm  •=.  the  mean  absolute  pressure,  being  such  an  ideal 
pressure  as  would  if   exerted  throughout  the 
stroke  produce  the  same  work  as  that  of  the 
variable  pressures, 
p6  =  the  mean  effective  pressure. 
The  equation  of  £  C  will  be,  page  103, 

2>  v  =  pl  v,  —  p  rl  =  pi  T,  (162) 


[HO.]  ISOTHERMAL    EXPANSION.          ,  173 

Saturated  steam  after  being  generated  in  a  boiler  is  con- 
ceived to  be  superheated  in  a  separate  vessel. 
We  have 

area  G-  B  C  II  '  =    /    '  p  dv  =  p,  vt  log*  —  ,         (163) 
JVi  0, 

and 

0^5  tf//  =  ^iV|  .(!  +  %.  r); 

also, 

pmv,  =  0  A  B 


Pi  r 

P*=Pm-P*>  (166) 

The  .effective   energy   exerted   by   one  pound   of   steam 
against  the  piston 

=  AB  GEFA  =  U=(pm  -  j93)  *v          (167) 

To  find  the  heat  absorbed  per  pound  of  steam,  let  H  = 
the  heat  absorbed. 

It  may  be  represented  by  a  diagram  thus  :  Let  L  be  the 
initial  state  of  the  feed  water 
as  to  pressure  and  tempera- 
ture, Tt  ;  A  the  state  at  the 
temperature  of  boiling,  T^ 
JIT  the  state  of  dry  saturated 
vapor  at  the  pressure  p1  and 
temperature  T^  ;  B  the  state 
of  the  superheated  steam  at  FIG-  50- 

the  pressure  pl  and  temperature  T6  ;  B  C  the  isothermal 
expansion  curve,  then  will  the  temperature  at  C  be  T^  the 
same  as  at  B.  (The  temperatures  marked  T9  and  T3  are  for 
use  in  the  two  following  exercises.) 


174  HEAT   ENGINES.  [HO.] 

Through  LAMB  and  C  draw  adiabatics,  then  H1  = 
<p,  L  A  <p,  =  C  (T,  -  T,)  =  the  heat  absorbed  by  the 
liquid  before  boiling  =  778  (T,  —  Tt} ;  Ht  =  £Te  =  <p, 
AM  <p,=  1117830  —  544.6  r,,  (Eq.  [78]  ) ;  //8  =  <p,  MB 
<PI  =  Kv  (Tt  —  T7,)  =  the  heat  absorbed  in  producing  su- 
perheating =  0.48  X  778  (Tt  -  Ts)  —  373  T6  -  373  Tt  - 
171953 ;  Jf4  =  (p4  B  C  <pb  =  the  latent  heat  of  expansion, 
which  equals  the  work  done  during  isothermal  expansion  of 
a  gas  considered  perfect  =  pl  vl  log  r,  Equations  (36), 
(163),  which  by  (164)  becomes 


From  equation  (162) 

373  T6  =  -typi  vl  nearly. 

.  •  .  H  =  //,  +  //,  +  //,  +  11,  =  9l  FA  B  C  <?„ 
=  694816- 


EXERCISE. 

1.  Let  p,  =   100   X   144  =  14400,   Tt  =  450°  r  =  10, 
?,  =  2i  X  144  =  360  Ibs. ;  2\  =  110°. 
Find  T,  =  327.6°, 

TB  =  910.66°, 
pl  v,  =  "75922  ; 

v,  =  5.27 ; 
rv,  =  52.7; 


p*  =  4396 ; 
U  =  231757; 

which  is  the  effective  work  done  by  one  pound  of  the  steam 
against  the  piston  ;  then,  (171), 

H  =  1132730  ft.-lbs., 

which  is  the  heat  expended  per  pound  of  steam   in  the 
cylinder. 


[111.]  ADIABATIC   EXPANSION.  175 

Pressure  equivalent  to  that  heat — 

p^  =  M  =  21364  Ibs., 

which  is  such  an  ideal  pressure  that  if  it  worked  against  the 
piston  while  it  swept  through  the  same  volume  as  when 
driven  by  the  one  pound  of  steam,  it  would  do  an  amount 
of  work  equal  to  the  entire  energy  of  the  heat  expended. 
Efficiency  of  the  steam — 


111.  Adiabatic  expansion  of  dry  saturated  steam. 
First,  assume  the  approximate  law 

p  v     =  p1vl    —  constant.  (172) 

The  work  during  expansion  will  be,  Fig.  49, 

G  B  C II  =  J£*p  d  v  =  p,  vt  (9  -  9  r-l\       (173) 
and  the  total  work  per  pound, 

OABCH^p.v,  (w  -  9r-A 
Terminal  pressure,  equation  (172) — 

*  =  £?•  (174) 

Mean  total  forward  pressure — 
O  A  B  C  H 


v* 
Mean  effective  pressure — 

/10      9    \ 

Work  done  per  POUND  of  steam — 


176  HEAT   ENGINES.  [1H-J 

Work  done  per  CUBIC  FOOT  of  steam  admitted — 

—  =  rpK  =  p,  (lO  —  9  r~±]  -  rpy    (178) 


Heat  expended  per  pound  of  steam  admitted  — 

This  will  be  the  heat  supplied  to  the  water  per  pound 

above  the  temperature  of  the  feed  water  plus  the  latent  heat 

of  evaporation,  and   is   given  by  equation  (93),  which  in 

the  present  notation  becomes 

H  =  e/(r,-T4)  +  //.,. 

Eq.  (78),         =  778(7;-7>f  837003-544.677,.  (179) 

Heat  expended  per  cubic  foot  of  steam  admitted  — 


i  (Ti  _  Tt)  +  L,  (Art.  78).        (180) 
Efficiency  of  the  steam  — 

E-.  (181) 

H 

EXERCISE. 

Let  p,  =  14400  Ibs.  ;  r  =  10  ;pt  =  360  Ibs.  ;  feed  water, 
110°  F.,  as  in  the  preceding  exercise. 

Then,  omitting  fractions  of  temperature  after  r,, 

ra  equation    (80),  =  788.26°,  using^,;  .-.  7\  =  327.60°. 
p*        «        (174),  =  1115  Ibs. 
r,,        "          (80),  =  640°,  nringjp,  ;     .-.  T,  =  180°. 
r,,        «          (80),  =  590°,     «     ^8;     /.  T,  =  134°. 

r4  =  110°. 

va  «          (89),  =  4.36  cu.  ft, 

vtt  «          (86),  =  4.37    "    " 

»r  "  (172),  =  43.7,  or.  100,. 

^  "  (175),  =  4363  Ibs.  per  sq.  ft. 

pn  u  (176),  =  4003    "      "     "     " 

U<  "  (177),  =  174931  ft-lbs. 

H,  "  (179),  =  85T706  "    " 


[112.]  EXPANSION   OF   SATURATED    STEAM.  177 

Efficiency  — 

%=  =  0.204.  (181o) 

J± 

112.  Adiabatic  expansion  of  saturated  steam 

according  to  the  theoretical  law.  If  the  steam  initially  have 
oe1  part  of  moisture,  use  equation  (&)  or  (ni),  p.  192.  If 
initially  dry,  Fig.  49  and  equations  (150)  and  (86)  give 


=   f 

J, 


p, 

=  J  [r>  -  r,  (l  +  log.  ^)]  +  ^-=p  Hw  (182) 
For  the  work  per  pound  of  steam  working  full  cycle, 
U=  A  B  CEF  =  J^  -  r2  (l  +  log.  ^)]  _L 

^^  H«  +  (p.  ~  P*}  uv  (183) 

The  heat  expended  per  pound  of  steam  admitted  to  the 
cylinder  will  be  the  same  as  in  the  preceding  Article,  or 

K  =  J(T1-  T)  +  ffw  (184) 

The  efficiency  will  be 


[Messrs.  Gantt  and  Maury  determined  the  Efficiency  of  Fluid  Vapor 
Engines  according  to  this  hypothesis  —  using  these  equations—  for  Water, 
Alcohol,  Ether,  Bisulphide  of  Carbon  and  Chloroform  (Thesis,  Ste- 
vens Institute  of  Technology,  1884  ;  Van  Nostrand's  Engineering  Mag- 
azine, 1884  (2),  pp.  413-432)]. 


EXERCISE. 

=  14400  Ibs.  ;  p3  =  360  Ibs.  ;  T,  =  110°  F.,  as  in 
the  preceding  exercise,  and  p^  =  1115  Ibs.,  as  found  in  that 


178  HEAT   ENGINES.  [112.] 

If  the  ratio  of  expansion  were  given,  pt  could  be  found 
only  by  a  tedious  approximation  ;  therefore,  we   have  as- 
signed the  final  pressure. 
We  have, 

rv  =  788.26°  ;  .  •  .  71,  =  327.66°,  as  before. 
r,  =  640°;        .-.T9=  180°,       " 
r.  =  590°  ;        .'.21=  134°,       « 
rt  =  570°  ;        .  •  .  T4  =  110°,       " 
j9,  =  14400  Ibs.  "         '* 

pt  =  1115  Ibs.  "        " 

d,  =  4.375  cu.  ft.  "         " 

q'  (88)> 


«  J  +        )  =  9.55,  Eq.  (152), 

17,    JJej  \  T,  T,    / 

Wj  =  ^  (778  hg.  ^  +  ^),  ^  (150), 

-*Je3     \  *i  Ti   ' 

=  rv,  =  41.76, 
0,  —  -w,  =  6.64. 
U  =  171507  ft-lbs.,  Eq.  (183). 
The  preceding  exercise  gives, 

H  =  857706  ft.  -Ibs. 
Efficiency  of  fluid— 


Steam  condensed  due  to  expansion  only — 

v-^r  =  °-137' 

or  nearly  14  per  cent. 
'est 
U  171507 


[112.]  EXPANSION   OF   SATURATED    STEAM.  179 

Mean  total  forward  pi^essure — 

j>ra  =  4170  +  360  =  4530  Ibs. 

It  will  be  seen  that  there  is  little  or  no  advantage  in  using 
the  exact,  but  more  complex,  formulas  of  this  Article  over 
the  approximate  ones  of  the  preceding  Article. 

The  efficiencies  found  in  the  three  preceding  cases  are  : — 

For  superheated  steam,  expanding  isothermally  (Ilia)  0.2C5 

For  saturated  steam,  expanding  adiabatically,  approximate  law 

(181a)  0.204 
theoretical  law 

(185a)  0.200 

The  effect  on  the  efficiency  by  superheating  is  too  small 
to  be  of  practical  importance.  As  this  fact  appears  to 
be  contrary  to  the  popular  opinion,  it  is  well  to  observe 
that  the  superheated  steam  in  Article  109  is  not  used  in  the 
most  economical  manner  ;  for  a  much  larger  amount  of  heat 
is  thrown  away  at  the  end  of  the  stroke  than  in  the  example 
of  saturated  steam,  so  that  if  it  were  utilized  in  heating 
feed  water,  or  worked  in  another  engine,  or  used  for  any 
other  useful  purpose,  the  efficiency  of  the  plant  would  be  in- 
creased. Or  if  it  had  been  expanded  down  to  that  of  the 
terminal  pressure  of  the  other  cases,  pt  =  1115  Ibs.,  it 
would  have  shown  a  greater  efficiency  ;  but  to  accomplish 
this  result  the  ratio  of  expansion  must  be  greater,  other 
data  being  the  same.  These  considerations  have  reference 
to  the  efficiency  of  the  fluid  only,  but  in  considering  the 
efficiency  of  the  plant,  the  size  and  cost  of  the  engine  enter 
as  elements  of  the  problem.  Thus,  to  do  the  respective 
works,  231757  and  174931,  deduced'in  two  of  the  preceding 
exercises,  with  two  engines  making  the  same  number  of 
revolutions  in  the  same  time,  according  to  the  conditions 
assumed,  the  volume  of  the  cylinder  of  the  one  supplied 
with  superheated  steam  must  be  larger  than  that  supplied 
with  saturated  steam  in  the  ratio  of  the  volume  of  a  pound 


180  HEAT   ENGINES.  [113.] 

of  superheated  steam  at  admission  to  that  per  pound  of  sat- 
urated steam,  or,  as 

5'2T    -120- 
£375  " 

but  the  ratio  of  the  works  done  will  be 

231757 

174931  - 

hence,  per  cubic  foot  of  the  cylinder  capacities  the  former 
engine  will  do 

£"  =  1.10  time. 

the  work  of  the  latter. 

The  engine  using  isothermal  expansion  and  doing  231757 
foot-pounds  of  work  per  pound  of  steam,  if  it  uses  tlu? 
pound  per  minute,  will  do 


33000 

horse-powers  per  pound  of  steam  ;  and,  per  hour,  it  will 
require 

1980000 
__=:b.o4  pounds 

per  horse-power.  The  engine  which  expands  adiabatically, 
doing  174:931  foot-pounds  of  work,  would  require 

1980000 

-iiiwr  =   11J"  po"lkKi 

per  horse-power  per  hour.  These  results  are  for  perfect  con- 
ditions, no  allowance  having  been  made  for  wastes,  clearance, 
or  initial  condensation  of  steam.  It  is  a  very  good  plant  that 
does  not  consume  more  than  seventeen  pounds  of  feed 
water  per  indicated  horse-power  per  hour,  although  re- 
liable records  of  some  good  tests  show  less  than  this  amount. 
Some  multiple  expansion  engines  have  been  reported  as 


[112-]  NO    EXPANSION.  181 

consuming  about  thirteen  pounds,  as  determined  from  the 
indicator  card,  but  that  mode  of  determining  the  weight  of 
steam  does  not  allow  for  the  condensation  of  steam.  The 
only  reliable  way  is  to  weigh  the  water  used.  Thirty  to 
forty  pounds  is  more  common  in  practice. 

The  heat  of  combustion  of  a  pound  of  pure  carbon  is  14500 
B.T.U.,  and  if  it  could  all  be  utilized  for  the  purpose  it  would 
evaporate  14500  -j-  966  =  15  pounds  of  water  at  and  from 
212°;  hence,  if  the  feed  water  be  at  110°  F.  and  boiling 
point  at  327°  F.,  as  in  the  two  preceding  exercises,  it  would, 
according  to  the  table  on  page  112,  evaporate  15  -~  1.14  = 
13.15  pounds  ;  and  to  develop  one  I.H.P.  per  hour  it  would 
require 

11.32  •*•  13.15  =  0.861  pounds 

of  coal.  This  does  not  allow  for  waste  in  producing  steam. 
If  the  efficiency  of  the  furnace  be  0.70,  it  would  require 
0.861  -7-  0.70  =  1.31  pounds  of  coal. 

Case  of  no  expansion.  In  many  simple  direct-act- 
ing steam  pumps,  the  full  pressure  of  steam  is  maintained 
throughout  the  stroke.  For  this  case  r  =  1  in  equation 
(177),  and  the  indicated  work  will  be 

&=  fa  -?,)*»  (185&) 

when  vl  is  the  volume  of  a  pound  of  the  vapor  at  the  pres- 
sure pr  The  work  done  during  the  forward  pressure  will 
be  the  external  work  performed  during  evaporation  at  the 
pressure^,,  and  is  sometimes  called  the  external  latent  heat  of 
vaporization.  That  part  of  the  apparent  latent  heat  which 
performs  disgregation  work  will  be  lost  at  the  exhaust. 

The  volume  of  the  cylinder,  the  piston  making  n  single 
strokes  per  minute  for  m  horse-powers,  will  be 


cu.ft.  (185.) 


182  HEAT   ENGINES.  [113.] 

"Water  consumed  per  indicated  horse-power  per  hour, 

jrr       33000  X  60 

—  ~  --  pounds. 


EXERCISES. 

1.  In  a  direct-acting  steam  pump,  let  the  uniform  gauge 
pressure  be  70  pounds  and  back  pressure  16  pounds,  feed  water 
60°  F.  ;  required  the  work  done  per  pound  of  steam,  effi- 
ciency, volume  of  the  cylinder  for  one  horse-power  if  there 
be  50  double  strokes  per  minute,  and  the  water  consumed. 

jp,  =  70  +  14.7  =  84.7, 

Pi-  P*  =  68-7, 

rl  =  775.6,  Eq.  (81)  ;  or,  Tt  =  315°  F.  ; 
v,  =  5.14,  Eq.  (86)  or  (89)  ; 

U  -  68.7  X  144  X  5.14  =  50849  ft.  lbs.,Eq.  (1855)  ; 
W  =  38.9  Ibs.  per  hour,  Eq.  (lS5d)  ; 
V  =  ^  Cu.  ft.  =  52.4  cu.  in.,  Eq.  (185c)  ; 
H  =  893844,  Eq.  (179)  ; 
E  =  0.057,  Eq.  (185), 

or,  the  theoretical  efficiency  of  the  fluid  will  be  about  5.7 
per  cent. 

In  actual  practice  the  loss  from  condensation  and  radi- 
ation in  these  pumps  is  considerable,  and  the  clearance  is 
not  only  relatively  large  but  somewhat  irregular—  especially 
in  the  smaller  sizes,  and  it  is  found  that  the  water  consump- 
tion ranges  from  75  to  125  pounds  per  horse-power  per 
hour,  with  the  possibility  of  being  outside  these  limits  in 
either  direction.  The  mean  average  efficiency  for  ilie  fluid 
admitted  (steam  and  water)  will  be  for  small  pumps  of  this 
class  about  -J  of  the  theoretical,  or, 

E  •=•  0.019,  approximately, 
and,     U  —  17000  ft.  Ibs.,  approximately, 
also,    W  =  120  Ibs.  of  water  per  horse-power  per  hour. 


[112.J  NO    EXPANSION.  183 

But  the  size  of  the  cylinder  need  not  be  correspondingly 
increased,  for  the  condensed  steam  will  occupy  but  little 
volume. 

The  efficiency  of  the  furnace,  boiler  and  connections  may 
be  taken  at  50  per  cent,  giving  for  the  entire  plant 

E'  =  0.0095, 

or  about  1  per  cent  of  the  theoretical  heat  of  the  fuel  burned 
in  the  furnace. 

It  has  been  found  .by  actual  measurements  that  the  average 
duty  (or  the  work  which  100  pounds  of  coal  can  do)  in 
direct-acting  pumps  feeding  75  to  100  horse-power  boilers, 
with  coal  of  good  quality,  may,  in  the  absence  of  direct 
experiment,  be  taken  as  10000000  foot-pounds.  This  is 
100000  foot-pounds  per  pound  of  coal,  or  100000  -f-  778  = 
128.5  thermal  units,  which  is  about  T^¥  of  the  heat  of  com- 
bustion of  the  average  of  commerical  coal.  The  efficiency 
of  such  a  plant,  then,  is  actually  about  1  per  cent  of  the  heat 
in  coal  of  good  quality.  Such  a  plant  will  require  from 
9  to  15  pounds  of  coal  per  indicated  horse-power  per 
hour. 

2.  In  the  preceding  Exercise,  if  the  stroke  be  five  inches, 
what  will  be  the  diameter  of  the  cylinder  ? 

3.  If,  in  a  direct-acting  steam  pump,  the  gauge  pressure 
be  100  pounds,  back  pressure  16  pounds,  feed  water  90°  F., 
find  the  efficiency  of  the  fluid,  and  compare  the  result  with 
that  in  Exercise  1. 

4.  If,  in  Exercise  3,  the  gauge  pressure  be  40  pounds, 
required  the  efficiency  of  the  fluid. 

5.  Explain  the  several  causes  of  the  loss  of  the  99  per 
cent  (more  or  less)  of  the  heat  of  combustion  as  found  in 
these  Exercises.     What  effect  has  the  temperature  of  the 
feed  water  upon  the  efficiency  ? 


184  HEAT   ENGINES.  [112a.] 

1 12a.    General  equations  of  vapor  engines. 

— Consider  only  one  pound  of  fluid  in  the  cylinder,  and  let 
B  C  be  the  curve  of  saturation, 
and  E  F  any  adiabatic  in  which 
there  is  only  a  fraction  of  the 
pound  of  vapor  throughout  the 
expansion. 

A  B,  Fig.  50«,  will  represent 

the  volume  of  a  pound  of  vapor  at 
FIG.  50«. 

the  absolute  pressure  O  A  =  p, 

and  absolute  temperature  r,,  G  /the  volume  at  the  absolute 

temperature  r  and  pressure  O  G  =  p. 

Let  xt  =  A  E  +-  A  B  =  the  fractional  part  of  the  fluid  at 

the  state  Et\\&i  is  vaporized, 
v,  =  A  B,  *,  vt  =  A  E, 
x  =  G  II  -r-  G  /, 

v  =  G  /,   volume  corresponding  to  the  pressure  O  G, 
x  v  =  G  II, 

c,  the  specific  heat  of  the  liquid, 

Ae,  the  latent  heat  of  evaporation  at  temperature  r  in  ordi- 
nary heat  units,  which  will  be 
Aei  at  temperature  rr 

Then   will   the  equation  of  the  adiabatic  E  F  be,  Eqs. 
(86)  and  (149), 

GH=  xv  =  (cloge  -IL+^'A)  ™,  .(«) 

^  f  ^  \  e 

which  may  be  put  under  the  more  symmetrical  form 

xhe  ,       r         x.  he.  T, 

— -  +  c  logo —  =  - — -  +  G  Iog6  — !-  =  a  constant,       (fy 

in  which  r0  is  any  arbitrary  temperature.  Since  the  vapor 
is  to  be  continually  saturated,  this  equation  is  limited  to  the 
conditions  that  a?,  must  not  be  negative,  and  must  be  less 
than  unity,  and  at  the  same  time  a?,  for  any  amount  of 
expansion,  must  be  less  than  unity. 


[112«.]  VAPOK   ENGINES.  185 

Let  subscript  „  be  used  for  the  terminal  state  F,  then 

a?QAe,          7        r,        as.h6.  T. 

-^  +  c  log*  -±  =  -±-£  +  G  log,— 

The  difference  between  the  initial  and  terminal  weights 
of  vapor  will  be 


and  this  may  be  negative,  zero,  or  positive.  We  will 
designate  those  vapors  whose  specific  heats  are  negative  as 
"  steam-like  vapors,"  and  those  which  are  positive  as  "  ether- 
like  vapors,"  steam  and  ether  being  typical  of  their  respec- 
tive classes. 

If  the  fluid  be  water,  then  c  =  1,  and  let  x,  =  0.436  at 
T  =  800°  F.  (absolute),  Ae  =  1436.8  -  0.7  r.  Then  equa- 
tion (a)  gives 

for  T  =  900°,  x  =  0.404,  t  =  600,  as  =  0.450, 

r  =  800°,  x  =  0.436,  T  =  500,  x  =  0.436, 

T  =--  700°,  x  —  0.450,  T  —  400,  x  =  0.407, 

T  =  650,  *  =  0.453,  T  =  200,  a?  =  0.277  ; 

from  which  it  appears  that  steam  increased  with  the  expan- 
sion as  the  temperature  fell  from  900°  to  650°,  or  from  340° 
to  190°  on  the  Fahrenheit  scale  ;  and  after  that  it  decreased 
continually  with  the  temperature.  This  change  of  the 
weight  of  steam  can  take  place  only  by  the  evaporation  of 
water  initially  in  the  presence  of  the  vapor,  and  by  condensa- 
tion later  in  the  expansion.  The  converse  is  also  true,  that 
if,  in  the  initial  state,  only  a  fraction  of  the  fluid  be  vapor, 
the  liquid  may  at  first  be  evaporated  by  adiabatic  compres- 
sion, but  it  may  reach  a  state  beyond  which  it  will  be  con- 
densed by  adiabatic  compression.  Thus,  in  the  example 
above  given,  if  at  600°  F.  (absolute)  45  per  cent,  of  the  fluid 
be  vapor,  it  will  increase  to  45.3  per  cent.,  after  which  it 
will  condense  indefinitely  with  adiabatic  compression. 


186  HEAT   ENGINES.  [112aj. 

If  at  650°  there  be  45.3  per  cent,  of  steam,  the  vapor  will 
condense  both  by  adiabatic  compression  and  expansion  from 
that  state. 

This  may  be  illustrated  by  the  annexed  diagram,  Fig. 
50£,  in  which  the  relations  are 
greatly  exaggerated.  Let  D  E  F 
represent  successive  states  of  con- 
stant steam  weight,  and  A  B  C  an 
adiabatic  of  part  liquid  and  vapor. 
These  curves  may  intersect  each 
other  at  two  points  a  and  I ;  above 
a  the  weight  of  vapor  in  the  adi- 
batic  will  be  less  than  at  «,  and 
the  adiabatic  will  lie  to  the  left 

of  D  E  F,  and  below  I  it  will  lie  below  the  curve  of  con- 
stant steam  weight.  The  adiabatic  is  less  curved  than  the 
curve  of  constant  steam  weight. 

To  find  the  minimum  weight  of  vapor  such  that,  by  con- 
tinued compression  of  steam-like  vapor,  the  liquid  will  le 
continually  evaporated. 

In  equation  (b)  first  find  the  value  of  r  that  will  make  the 
left  member  a  minimum  when  x  =  1.  Neglecting  all 
powers  of  r  above  the  first  in  the  latent  heat  of  evaporation, 
Regnault's  experiments  give 

Ae          a        , 

—  =  --*• 

where  a  and  £  are  constants  depending  upon  the  particular 
fluid.  Using  this  value,  it  will  be  found  that  the  required 
function  is  a  minimum  for 

a 


that  is,  r  will  be  near  the  "  temperature  of  inversion," 
which,  in  the  case  of  steam,  is  about  1436°  F.  (absolute),  or 
976°  F.  actual.  Since  the  law  of  the  latent  heat  of  evapo- 


tH2«.]  VAPOR  ENGINES.  187 


ration  here  given  is  not  exact,  ;  nd,  even  if  it  were, 
naulfs  experiments  would  not  warrant  the  extension  to 
such  high  temperatures,  we  will  discard  fractions,  and 
treat  the  entire  number,  1436,  as  if  it  were  exact.  Since 
the  adiabatic  law  is  not  applicable  above  this  state,  the 
maximum  condensation  by  adiabatic  expansion  will  be 
found  by  beginning  at  this  state  and  expanding  down  to 
the  required  temperature.  In  equation  (<:•),  letting  x^  —  1, 
TI  .-  1436,  7<ei  =  1436—0.7  T,  c  =  1,  then 


: 14-36 


2.3(>26%10  —  -  .+  0.3 
1  —  x  =  1  —  - 


1436  _„ 


Abso.  Temp.   Per  cent,  of  Steam.  Per  cent,  of  Water.         Temp.  Deg.  F. 

If  r  =  800,  x  =  0.808,  1  -  x  =  0.192,  340. 

=  TOO,  x  =  0.753,  1  -  x  =  0.247,  240. 

=  672,  x  =  0.725,  1  -  x  =  0.265,  212. 

=  600,  x  =  0.692,  1  —  x  =  0.308,  140. 

It  thus  appears  that  if  72£  per  cent,  of  the  fluid  be  satu- 
rated steam,  or  26^  percent,  of  it  be  water  at  212°  F.,  the 
steam  will  condense  continually  by  adiabatic  expansion,  or 
the  water  be  continually  evaporated  by  adiabatic  compres- 
sion. If  there  be  less  than  twenty-six  per  cent,  of  water  at 
212°,  the  water  will  all  become  evaporated  before  the  tem- 
perature reaches  the  critical  temperature,  and,  after  passing- 
that  state,  compression  will  produce  superheating.  Every 
adiabatic  having  more  than  72£  per  cent,  of  steam  at  212° 
is  tangent  to  some  curve  of  constant  steam  weight ;  and 
hence,  with  the  exception  of  the  adiabatic  tangent  to  the 
curve  of  saturation,  will  have  a  state  of  maximum  steam 
weight,  at  which  point  the  curves  of  constant  steam  weight 
and  the  adiabatic  will  have  a  common  tangent.  From  this 
state  condensation  of  steam  will  result  from  compression  as 
well  as  from  expansion.  The  adiabatic  which  is  tangent  to 


188 


HEAT    ENGINES. 


the  curve  of  saturation  passes  through  the  state  of  the  tem- 
perature of  inversion. 

According  to  the  preceding  table,  at  T  =  672°,  if  72£  per 
cent,  is  steam,  compression  will  produce  evaporation  up  to 
1436°.  If  at  r  =  672°  we  assume  70  per  cent,  of  steam,  we 
find  the  following  results  : 


T  =  672, 

a;  —  0.70. 

r  =  1100, 

x  —  o.849. 

T  =  700, 

x  =  0.73. 

T  =  1200, 

x  =  0.855. 

r  =  800, 

a?  =  0.76. 

T  =  1250, 

x  =  0.860. 

r  =  900, 

x  =  0.794. 

T  =  1300, 

a?  =  0.859. 

T  =  1000, 

x  =  0  825. 

T  =  1400, 

x  —  0.84. 

It  will  be  seen  that  the  amount  of  steam  will  be  a  maxi- 
mum at  1250°  F.  absolute. 


PIG.  50c. 

In.  Fig.  50c  the  dotted  lines  are  curves  of  equal  steam 
weights,  and  the  full  lines — except  the  curve  of  saturation — 
are  adiabatics,  one  of  which  is  tangent  to  the  curve  of  satura- 
tion ;  another  tangent  to  the  curve  whose  constant  steam 
weight  is  86  per  cent.,  the  point  of  tangency  being  at  the 


[112«.]  VAPOR    ENGINES.  189 

temperature  of  790°  F.  ;  another  is  tangent  to  the  curve  of 
50  per  cent,  of  steam  at  240°  F. ;  and  the  fourth  tangent  to 
the  curve  of  45.3  per  cent,  of  steam  at  190°  F.  absolute. 

In  order  to  show  the  properties  on  a  small  scale,  it  is 
necessary  to  exaggerate  the  relations,  thus  distorting  what 
would  be  the  correct  figure. 

An  examination  of  ether  will  show  that  the  results  here 
deduced  for  steam  are  not  necessarily  applicable  to  other 
vapors.  In  "  ether-like  vapors  "  the  temperature  of  inver- 
sion is  below  ordinary  temperatures  ;  and  for  such  if  a?,  =  1, 
condensation  will  result  from  adiabatic  compression  for 
temperatures  above  that  of  inversion.  Thus,  for  ether, 
omitting  terms  above  the  first  power  of  r,  we  have  from 
Regnault'  s  experiments, 

7>e  =  93.3214  +  0.3870  r. 
c  =:    0.517. 

Hence,  from  equation  (139),  page  147, 


s  =  0.517  —         —  =  specific  heat  of  the  saturated  vapor. 

If  s  —  0,  then  r  =  180°  (absolute),  or  —  280°  F.  ;  and 
this  is  the  temperature  of  inversion.  Assuming  any  tem- 
perature above  this,  as  r^  =  520°,  and  a?,  =  1  in  equation 
(a),  then 

0.5664  -  2.3026  %10-^ 

"~       ^  +  0.3870      ' 

From  this  it  appears  that  x  will  diminish  as  r  increases, 
and  finally  become  zero  for  r  =  915°,  nearly. 

There  appears  to  be  no  proportion  of  vapor  to  liquid  such 
that  they  will  be  the  same  at  two  different  states  on  an  adia- 
batic,  as  has  been  found  for  steam.  It  may  be  shown  that 
for  any  value  of  »„  x  will  decrease  as  r  increases,  showing 


190  HEAT   ENGINES.  [  113ft.] 

that  reevaporation  does  not  take  place  during  adiabatic  com- 
pression. 

If  the  fluid  be  initially  all  liquid,  then  xl  =  0,  which  in 
equation  (a)  gives  for  the  equation  of  A  J,  Fig.  50#, 

TV  ,       TI 
xv  =  c-j-  Iog9  -,  («) 


This  expression  may  in  some  cases  have  a  maximum, 
from  which  it"  appears  that  if  the  fluid  be  initially  all  liquid, 
under  adiabatic  expansion  the  liquid  may  be  evaporated 
until  the  temperature  is  so  reduced  as  to  produce  the  maxi- 
mum weight  of  vapor,  after  which  the  vapor  will  condense. 

Thus,  for  steam  <•  =  1,  and  if  r,  =  800,  x  will  be  a  maxi- 
mum for  T  =  350°  (absolute),  nearly,  at  which  state  x  will 
be  0.21 ;  or  2-t  per  cent,  of  the  liquid  will  have  become 
vapor.  At  300°,  x  =  0.239 ;  for  r  =  200°,  x  =  0.21.  All 
these  latter  temperatures  are,  howeveit,  so  much  below  any 
used  in  practice,  that  it  is  not  probable  that  the  formula  for 
evaporation  will  be  applicable ;  and  we  may  assert,  that, 
within  practical  limits,  steam  will  be  continually  generated 
under  adiabatic  expansion,  if  in  the  initial  state  the  fluid  be 
entirely  water. 

With  ether,  if  initially  liquid,  evaporation  will  increase 
with  adiabatic  expansion  until  it  all  becomes  saturated  vapor, 
after  which  it  will  superheat ;  provided  that  the  liquid  be- 
comes vapor  before  the  temperature  of  inversion  is  reached. 

The  numerical  values  of  these  results  will  be  modified  in 
some  cases  considerably — if  higher  powers  of  the  temper- 
ature be  included  in  the  analysis. 

The  ratio  of  expansion  will  be 


[112a-]  VAPOK   ENGINES.  191 

If,  at.  the  cut-off,  B,  tlie  fluid  be  all  vapor,  as  it  may  be 
for  steam-like  vapors,  then  a?,  =  1,  and  reducing  by  means 
of  equation  (a)  we  have 


which  is  the  equivalent  of  equation  (152),  page  154. 

For  ether-like  vapors,  if  the  final  state  is  that  of  vapor 
only,  then  a?2  =  1,  and  substituting  a?,  from  equation  (a) 
gives 

_  v,  A,.  . 


The  weight  of  ether  vapor  at  J5,  the  beginning  of  expan- 
sion, in  order  that  the  pound  of  fluid  shall  be  all  vapor  at 
C,  the  end  of  the  expansion,  will  be  »,  in  equation  (a)  when 
a?  =  1,  or 


In  practice,  the  adiabatic  expansion  of  steam-like  vapors 
may  be  approximately  realized,  but  there  is  well-nigh  an 
insuperable  difficulty  in  securing  the  adiabatic  expansion  of 
saturated  ether-like  vapors  ;  for,  in  the  former  case,  if  steam 
be  in  the  state  of  saturation  at  the  instant  of  the  cut-off,  it 
will  continue  to  be  saturated  during  expansion  ;  but,  with 
the  latter,  if  no  ether  liquid  be  present  at  the  instant  of  cut- 
off, the  vapor  will  superheat  during  expansion,  and  instead 
of  realizing  equation  (a),  the  curve  of  expansion  will  be  of 
the  form 

p  -yn  =  a  constant, 

in  which  n  will  be  the  ratio  of  the  specific  heat  at  constant- 
pressure  to  that  at  constant  volume.  We  will  continue  to 
consider  the  vapor  as  saturated. 


192  HEAT    ENGINES.  [112«.] 

To  find  the  work  A  E  F  D,  Fig.  50a,  p.  184,  the  expan- 
sion E  F '  being  adiabatic,  the  vapor  being  saturated 
throughout  expansion,  we  have 


Z7,  =  A  EFD  =GIL  dp  = 


=  J\<-  (r,  -r,-  r.loff 

which  becomes  equation  (182)  if  go,  =  1. 

If,  in  this  expression,  the  value  of  a?,  from  equation  (a)  be 
substituted,  and  subscript  „  be  attached  to  those  variables 
which  are  without  subscripts,  we  will  have 


,,]  ; 


Equation  (&)  is  better  adapted  for  the  discussion  of  steam- 
like  vapors,  and  equation  (I)  for  ether-like  vapors ;  for  in  the 
former  x,  may  be  unity,  and  in  the  latter  a?a  may  be  unity. 

Eliminating  log,.  — -  from  these  equations  by  means  of 
equation  (a)  gives 

U,  =  j[c  (r,  -  r,)  +  «,  A.,  -  a,  Ae,  ],  (m) 

in  which  a?,  and  a%  are  limited  as  before. 

If,  during  the  return  stroke,  the  fluid  be  refrigerated  so 
as  to  maintain  the  constant  temperature  rv  the  pressure  will 
be  uniform  and  equal  CD  ;  and  if  at  some  point,  as  </,  adia- 
batic compression  begins  and  is  continued  until  the  tempera- 
ture is  raised  to  r,  at  A,  let  a?,  be  the  weight  of  vapor  at 


[H2«.J  VAPOR   ENGINES.  193 

state  A,  then  will  the  work  done  by  compression  be  found 
by  simply  changing  xl  to  xn'  since  all  the  other  quantities 
remain  as  before  : 


'.'.  U.  =  J  [c  (r,  -r,-  rjoge  Ij)  +  IL=!'  x,  £.,];  (n 
hence  the  work  done  in  the  cycle  A  E  F  J  A  will  be 


The  heat  absorbed  will  be 

Jh-ei  (x,  -  aj.Jj  ; 
hence,  the  efficiency  will  be 


which  is  the  same  as  that  of  the  perfect  elementary  engine. 
Neglecting  compression  and  clearance,  we  have 

U=AEFD  f  (>t  -.?,)»,*>„ 

where  p^  =  0  J),  p3  =  0  M,  absolute  pressures.     If  rt  be 
the  temperature  of  the  feed  water,  the  heat  expended  will  be 

H  =  Jc  (r,  -  r^  +  x,  7/ei, 
where  Hei  =  Jhei.     Hence  the  efficiency  will  be 

J  L  c  (  TI  —  r»  —  r»  %  ^  )  +  T'  ~  ^  Xi  Tlei  \  +  (P*  —  Pa)  X*  v* 

E=  -  --    (0) 

J  [  c  (r,  -  r4)  +  a,  AelJ 

From  this  result  it  appears  that  in  the  case  of  actual 
engines,  the  specific  heat  of  the  working  fluid  and  the  latent 
heat  of  evaporation  both  affect  the  efficiency.  If  the  feed 


194  HEAT   ENGINES.  [112a 

water  be  at  the  temperature  of  the  exhaust,  then  r4  =  r9 
and  the  preceding  expression  may  be  reduced  to 


ft  -—  T,          c  r, 

E= 


r,  c  (r,  -  r,)  +  J,  hel 

By  retaining  ,r,  and  a1,,  equation  (o)  is  applicable  both  to 
"  steam-like"  and  "  ether-like""  vapors,  only  observing  that 
neither  xt  nor  a%  can  exceed  unity,  and  that  they  are  related 
to  each  other  through  equation  (a). 

To  find  the  work  done  during  adiabatic  expansion  when 
the  initial  state  A  is  that  of  liquid  only,  make  a?,  =  0  in  the 
value  of  Un  or  a?,  =  0  in  equation  (#),  giving 


=  ADJ=Jc 


and  if  the  temperature  at  /  be  rt  then  will  A  M  L  be  found 
by  substituting  rs  for  rt  in  the  preceding  equation. 

Actual  engines  do  not  expand  down  to  the  back  pressure, 
neither  is  the  pound  of  fluid  retained  in  the  cylinder  ;  but 
at  the  end  of  the  expansion  the  exhaust  is  opened,  and  the 
vapor  escapes  until  the  exhaust  is  closed  at  the  point  L  in 
the  back  stroke.  The  adiabatic  A  .L  will  then  be  for  only 
a  fraction  of  a  pound  of  fluid.  To  find  its  equation  let  z 
be  the  fraction  of  the  pound  of  fluid,  including  both  liquid 
and  vapor,  then  equation  (a)  gives 

T,  3C,    hK. 


If  the  fluid  be  all  liquid  at  A,  then  x,  =  0,  and 

,.         _       ??-.,^.3,     .  _  . 

which  reduces  to  equation  (f  ),  if  z  =  1  as  it  should. 


[H2«.]  VAPOK    ENGINES.  195 

But   in  practice  there  is  clearance  and  the  fluid  will  not 
be  reduced  to  a  liquid  at  state  A.     Representing  the  clear- 
ance by  P  A,  Fig.  50r7 ;  then  will  z 
= '  P  A  +•  A  £,  where  A  B  is  the  p    A      B 
volume  of  a  pound  ;  and  equation  (•/•) 
will  be  the  equation  of  the  adiabatic. 
It  will,  however,  be  more  convenient  Q 
to  use  the  approximate  equation  N 

p  vV  =  p,  vV, 

as  has  been  done   in   the    following 
equation  (it). 

Some  practical  considerations.  A  steam  or  hot- 
air  jacket  is  sometimes  used  to  prevent  liquefaction  of  steam 
in  the  cylinder  by  keeping  the  walls  of  the  cylinder  hot. 
Liquefaction  of  steam  in  the  jacket  produces  no  bad  effect, 
although  it  represents  cost  for  fuel.  The  entire  action  is  too 
complex  to  admit  of  definite  computation. 

The  velocity  of  the  steam  through  the  steam  pipe,  if  not 
more  than  100  feet  per  second,  does  not,  according  to  D.  K. 
Clark,  produce  any  appreciable  loss  by  frictional  resistance. 
The  loss  of  pressure  in  passing  through  the  ports  into  the 
cylinder  is,  in  practice,  from  3  to  10  pounds,  and  in  excep- 
tional cases  even  more. 

Wire-drawing  of  steam  is  the  reduction  of  pressure  due  to 
friction.  This  does  not  represent  a  corresponding  wraste  of 
energy,  for  it  produces  heat,  thus  superheating  it — that  is, 
produces  a  temperature  higher  than  the  boiling  point  corre- 
sponding to  its  pressure,  the  pressure  being  lower  than  at  the 
boiler  ;  but  the  entire  energy  is  never  restored  in  this  way. 
It  is  better  to  cut-off  earlier  with  throttle  open  than  to 
throttle  and  cut-off  later,  to  produce  the  same  work. 

Superheating  may  be  produced  by  wire-drawing,  by  a 
steam-jacket,  by  circulating  a  hot  fluid  through  flues  in  the 
steam-chest,  by  heating  the  pipes  conducting  the  steam  to  the 


196  HEAT   ENGINES.  [I12a.] 

engine,  by  heating  in  the  cylinder  by  hot  pipes,  or  by  inject- 
ing some  superheated  vapor  into  a  body  of  saturated  vapor. 
The  object  of  superheating  is  to  prevent  condensation,  to 
diminish  the  back  pressure  by  producing  steam  of  less  den- 
sity, and  to  increase  the  efficiency  of  the  fluid. 

When  steam  is  superheated  to  such  an  extent  that  it  may, 
without  material  error  in  practice,  be  treated  as  perfectly 
gaseous,  it  is  sometimes  called  steam  gas.  Experiments  of 
Him  and  others  show  that  a  very  moderate  amount  of  super- 
heating produces  steam  gas ;  from  which  it  is  inferred  that 
the  formulas  for  steam  gas  will  be  practically  correct  for 
ordinary  superheated  steam. 

EXERCISE. 

Find  the  work  per  pound  of  ether  working  in  an  engine 
without  clearance  or  compression,  expansion  complete,  be- 
tween the  absolute  pressures  of  100  and  14.7  pounds  per 
square  inch  ;  the  ratio  of  expansion  and  the  efficiency,  the 
fluid  being,  entirely  saturated  vapor  at  the  end  of  the  expan- 
sion, and  the  temperature  of  the  liquid  ether  60°. 

Since  expansion  is  complete,  the  final  pressure,  14.7  pounds, 
will  equal  the  back  pressure. 

The  specific  heat  of  liquid  ether  is  c  =  0.517.  To  find 
the  initial  and  terminal  temperatures  we  have,  equation  (81), 

1 


JA  -  loffup       &        B_, 

C  r4  C'1      26' 

in  which  for  ether,  A  =  7.5041. 

B  =  2057.8,  log  B  =  3.313425. 
C  =  104950,  lo'j  C  =  5.217355. 

Hence,  r,  =  670,  ra  =  558 ;  .  • .  T,  —  216°  F.,  T,  =  98°  F. 
Latent  heat  of  evaporation  as  determined  by  Regnault, 
A   =  171.24  —  0.0487  T  —  0.000473  T* 


L112«-l  VAPOR  ENGINES.  19? 

hence,  for  terminal  state,  since  T^  =    98°.  .  .  .h^  =.       16^5 

for  initial  state,  since  T,  =  216°  ____  hei  =       139 

Work,  ABCD,  Fig.  50a,  o?t  =  1  in  Eq.(Z),ft.-lbs.  U  =  22200 
Initial  weight  of  vapor,  Eq.  (/),  Ibs  ...........  a?,  =  .     0.92 

Efficiency,  Eq.  (o),  making  a?,  =  0.92,  p,  =  p3.  E  —     0.137 
Volume  of  a  Ib.  of  liquid  ether,  cu.  ft  ........  v  =  0.0223 

Vol.of  Ib.of  vapor  at  p,  =  100  lbs.,Eq.(84),cu.  ft.  vt  =  2.98 

"  ««  "  "  "^,  =  14.7"  "  "•  "  "  vt=  12.52 
Ratio  of  expansion,  Eq.  (/)  .................  r  =  4.6 

Tf  tfA0y«  &e  a  clearance,  and  sufficient  fluid  be  retained  to 
just  fill  the  clearance  by  compression,  as  indicated  by  L  A, 
Fig.  50^?,  this  fluid  will  act  as  a  cushion,  and  the  energy  in  it 
will  be  stored  and  restored  with  each  stroke,  and  will  not 
form  any  part  of  the  working  fluid.  In  this  case,  freeing 
the  diagram  of  the  effects  of  the  cushion  fluid,  as  in  Stir- 
ling's engine,  page  224,  ut  will  be  represented  by  a  line 
equal  to  J  C,  in  which  case  equation  (n)  becomes  ap- 
plicable by  subtracting  from  it  a  trilinear  area,  of  which 
the  hypothenuae  is  the  curved  line  J  L,  and  the  base  the 
projection  of  L  J  on  F  E.  But  the  mean  effective  pres- 
sure will  be  diminished  because  the  effective  work  per 
revolution  will  be  less,  the  back  pressure  being  greater. 

Ratio  of  expansion  with  clearance.  A  £  C  E  Z,  Fig. 
50J,  being  the  diagram  described  by  the  indicator,  the  cut-olf 
being  at  B,  A  B  will  be  the  apparent  steam  line,  and  P  B 
the  real  steam  line. 

Let 

F  E 

r'  —  —  ^  =  the  apparent  ratio  of  expansion, 

0  C 

r   =  --  =  the  real  ratio  of  expansion, 


PA 

c   =  ——  =  the  ratio  of  the  volume  of  clearance  to 

.  the  piston  displacement, 
s  —  F  E  =  the  stroke  of  the  piston  ; 
then 


198  HEAT    ENGINES. 


(fi 


The  adiabatic  A  L  will  be  for  a  fractional  part  of  one 
pound  of  the  vapor,  and  if  A  B  be  proportional  to  the 
volume  of  one  pound  of  the  vapor,  and  A  J  terminates  at 
the  end  of  the  clearance,  then  will  the  weight  of  vapor  re- 
quired to  produce  A  L  be 

AP 

of  one  pound  =  c  r  ; 


and  the  volumes  will  be  in  the  same  ratio  at  equal  pressures. 
Hence, 

N  L  _  c  r'  vt 

PA-^^/' 

"")*•  00 

which  determines  the  point  where  compression  must  begin. 
The  mean  effective  pressure  will  be  diminished,  but,  like 
many  other  elements,  the  exact  amount  cannot  be  deter- 
mined theoretically,  except  by  a  full  solution  of  the  problem  ; 
still  a  sufficiently  near  approximation  may  be  found  by 
means  of  equation  (176),  page  175,  which  is 


in  which  r*  will  be  between  1  and  2  for  all  ratios  of  expan- 
sion used  in  practice,  and  when  pt  is  large  compared  with 
Pv  P*  wiH  yary,  approximately,  inversely  as  r.  Hence,  if 
PS  be  the  mean  effective  pressure  with  clearance,  and  p* 


[112«.]  VAPOR  ENGINES.  199 

without,  we  have,  approximately, 

P*  =  ?  P*'  W 

The  piston  displacement  per  minute  in  doing  the  same 
work  as  without  clearance  will  be  increased  in  the  ratio 

r'  -r-  r. 

If  the  steam  is  completely  exhausted  during  each  return 
stroke,  the  real  volume  of  steam  will  be 

P  B  =  (1  -f  c  r')  A  £,  (w) 

or  1  -f-  c  r'  times  the  apparent  volume. 

The  mean  absolute  pressure  will  also  be  diminished,  to 
find  an  approximate  expression  for  which,  conceive  that  the 
piston  displacement  equalled  the  volume  of  the  cylinder, 
including  the  clearance  ;  then  would  the  work  done  be 

pm  (1  +  c}  u¥ 
But  the  work  done  in  passing  over  the  clearance  would  be 

PI  c u, ; 

and  if  pm'  be  the  mean  absolute  pressure,  we  have 
Pm  u*  =  pm  (1  +  c)  u,  —  p,  c  u, ; 

.•••  jv  -P*  =  C(PI  -  p^>  («o 

The  expenditure  of  heat  per  pound  of  steam  per  stroke 
without  clearance,  or  with  cushion  space  just  filled  by  the 
compression  of  vapor,  being 

H  =  JJei.+  <7(r,-72),  (y) 

with  clearance  and  complete  exhaustion  with  each  return 
stroke  will  be,  equations  (w)  and  (y\ 

H  (1  +cr')  =  [#„  +  C(r,  -.r,)]  (1  +  c  /).      (z) 
The  efficiency  of  the  fluid  will  also  be  diminished  /  for 

effective  work  with  clearance      _  pmf  — jp3t 
effective  work  without  clearance       pm  —  p3 ' 


200  HEAT   ENGINES.  [112a.] 

and  the  expenditure  of  steam,  and  consequently  of  heat,  is 
greater  with  clearance,  as  shown  by  equation  (3),  and 
hence  the  diminution  of  the  efficiency  will  be 

E-  E'  =  -.  --  2=4—  —  o  nearly. 


Although  the  effects  due  to  these  and  other  practical  con- 
siderations cannot  be  thoroughly  analyzed,  yet  their  dis- 
cussion shows  that  they  do  not  oppose,  or  revolutionize,  the 
general  theory  of  the  vapor  engine  —  they  simply  modify 
its  results.  A  more  complete  knowledge  of  vapor  engines 
requires  special  experiments  and  a  study  of  the  engine  itself 
under  varied  conditions.  Theory  teaches  much,  and  we  are 
thankful  that  we  know  so  much,  and  regretful  that  we  know 
so  little. 

113.  Cut-off,  With  a  given  plant,  if  the  cut-off  be 
early  more  work  may  be  done  with  a  given  amount  of  fuel 
than  if  the  cut-off  be  late  in  the  stroke  ;  and  it  is  proposed 
to  find  the  cut-off  which  shall  give  the  most  work  per  pound 
of  steam  admitted  to  the  cylinder.  This  problem  may  be 
called  the  point  of  'cut-off  'that  will  produce  the  greatest  Effi- 
ciency of  Fluid. 

With  a  given  plant,  it  may  be  proposed  to  find  the  point  of 
cut-off  such  that  the  owner  may  realize  the  greatest  profit 
by  selling  the  power  produced.  This  condition  will  involve 
the  first  cost  of  the  plant,  attendance,  repairs  and  deteriora- 
tion. The  deterioration  may  be  such  that  the  cost  of  the 
entire  plant  will  be  absorbed  in  the  course  of  a  few  years, 
or  if  sold  during  this  time,  it  will  be  the  difference  between 
the  original  cost  and  the  amount  received  by  the  sale.  In 
this  case,  if  the  cut-off  be  early  fuel  may  be  saved,  but  the 
other  changes  may  make  the  cost  of  the  power  delivered 
more  per  dollar  expended  than  if  the  cut-off  were  later,  thus 
making  it  a  problem  of  maxima  and  minima.  This  may  be 
called  the  Owners  Problem. 

Again,  in  the  plant  of  the  preceding  case,  the  parts  may 
be  improperly  proportioned  :  but  if  a  definite  amount  of 


[113.]  CUT-OFF.  201 

work  is  to  be  produced,  the  designer  may  be  required  to 
proportion  the  plant  so  that  the  boilers  shall  be  of  the  proper 
size  for  working  most  economically  for  producing  the  re- 
quired amount  of  steam,  and  the  engine  so  proportioned 
that  by  cutting  off  properly  the  power  produced  shall  cost 
the  least  per  dollar  expended.  If  cut-off  be  too  late  in  this 
case,  more  steam  will  be  required,  requiring  larger  boilers 
and  more  fuel,  while  the  engine  may  be  smaller,  thus  cost- 
ing less  ;  or  if  cut-off  be  too  early,  requiring  less  steam  and 
smaller  boilers,  the  cylinder  and  every  part  of  the  engine 
must  be  larger,  costing  more,  so  that  this  is  a  problem  of 
maxima  and  minima,  and  may  be  designated  as  the  Design- 
er's Problem. 

These  and  similar  problems  have  received  the  general 
title,  The  Most  Economical  Point  of  Cut-off. 

A  general  solution  of  the  owner's  problem  was  made  by 
Rankine,  and  is  made  the  basis  of  the  solution  of  the  other 
two.  It  is  substantially  as  follows : — 

Let  pl  =  the  initial  absolute  pressure  in  the  cylinder  per 

square  foot, 

pm  —  the  mean  absolute  pressure, 
F  =  the  resistance  of  the  engine  other  than  the  use- 
ful load,  including  friction  and  back  press- 
ure, 

h  =  the  cost  of  producing  unity  of  weight  of  steam 
in  unity  of  time  (one  hour),  which  consists  of 
the  cost  of  fuel,  repairs,  wages  of  firemen,  in- 
terest on  cost  of  boilers,  and  depreciation  ; 
&  =  interest  on  the  cost  of  the  engine,  plus  engi- 
neer's wages,  plus  cost  of  repairs,  plus  depre- 
ciation of  value  of  engine,  plus  cost  of  waste 
and  oil,  reduced  to  cost  per  square  foot  per 
hour ; 

A.  =  area  of  the  piston  in  square  feet. 
I  =  length  of  stroke  in  feet. 


)2  HEAT   ENGINES.  [113.] 

n  =  the  number  of  times  the  cylinder  takes  steam  in 
unity  of  time  (one  hour),  once  per  revolution 
for  single-acting  engines,  and  twice  for  dou- 
ble-acting ; 

I  n  =  the  number  of  linear  feet  swept  through  by  the 
piston  in  unity  of  time — one  hour  ; 

•y  =  the  volume  of  unity  of  weight  of  steam  at  the 
pressure  p^  taken  from  a  table  of  "  saturated 
steam  "  or  from  equation  (86),  or  (89) ; 

W  =  weight  of  steam  used  by  the  engine  in  unity  of 
time — one  hour, 

vt  =  the  volume  swept  through  by  the  piston  in  one 
stroke, 

i>,  =  the  volume  swept  through  to  the  point  of  cut- 
off, 

r  =  the  ratio  of  expansion  =  vt  -4-  vt 

Z  =  an  auxiliary  quantity  =  pm  v,  -r-  pt  vt  =  the 
ratio  of  the  work  done  at  full  stroke,  working 
expansively,  to  the  work  done  up  to  the  point 
of  cut-off. 
Then 

'   ass) 

.-.pm=*i  Z.  (187) 

Tc  A  =  the  interest  on  the  cost  of  the  engine  per  hour 
for  the  steam  used,  and 

*  ^  —  the  interest  for  each  pound  of  steam  used  per 
hour ;  hence,  the  total  cost  per  pound  per 
hour  will  be 


[113.]  CUT-OFF.  203 

The  volume  of  TF  pounds  of  steam  will  be  v  TF,  and  at 
full  stroke,  r  v  TF; 

.-.  Aln  =  rv  TF;  (188) 

and  the  preceding  expression  becomes 

z    t    7  rv 
*+'*** 

The  useful  work  per  stroke  will  be 

U=(pm-F)v» 
which  per  pound  of  steam  per  hour  becomes 


(189) 


which  by  means  of  Equations  (187)  and  (188)  becomes 


hence,  the  work  done  per  unit  of  cost  (one  dollar)  of  steam 
will  be 

Z-*Lr 

v(p,Z~Fr]  p. 

-— 


which  is  to  be  a  maximum  in  reference  to  r  as  a  variable, 
and  will  be  a  maximum  when  the  factor 

Z-  F-r 

TT7T 


is  a  maximum.  Equation  (186)  shows  that  Z  is  &  function 
of  pm  and  r  ;  but  the  form  of  the  function  pm  is  not  defi- 
nitely known,  depending,  as  has  been  previously  stated, 
upon  the  behavior  of  the  steam  in  the  cylinder  —  whether  it 
be  dry,  saturated,  superheated,  wire-drawn,  &c.  If  the 


204 


HEAT   ENGINES. 


[113.] 


mean  forward  pressure  be  found  in  any  manner  as  a  func- 
tion of  r,  a  graphical  solution  may  be  made  as  follows  : 
Draw  two  axes,  O  X  and  0  Y,  and 
construct  the  locus  J.  B,  Fig.  50,  rep- 
resenting Equation  (!$<!),  by   laying 
off  spaces   0  Q  on  O  X  to  represent 
r  and  corresponding  ordinates   Q  P 
to  represent  Z.     Through  the  origin 
0  draw  a  line  0  N,  such  that 
F 


FIG.  50. 


tan  NOX  = 


then  will  any  ordinate  Q  JV  be 


-which  is  the  numerator  of  expression  (191). 
On  the  negative  axis  of  x  lay  off  a  distance 


00  = 


kin 
TV 


and  at  C  erect  a  perpendicular  intersecting  N  O  prolonged, 
at  D.  From  D  draw  a  tangent  to  the  locus  A  12,  the  point  of 
tangency  being  P,  then  will  the  corresponding  abscissa,  0  Q, 
be  that  value  of  r,  which  will  make  (191)  a  maximum.  For, 
the  part  P  N  of  any  ordinate  between  D  JV  and  D  P  will 
be  proportional  to  C  Q,  its  distance  from  C ;  but  C  Q  is 
the  denominator  of  (191),  and  if  the  line  DP  be  above 
the  tangent,  nothing  will  be  determined  by  it,  and  if  below, 
assume  that  it  passes  through  some  point,  as  J.,  on  the 
curve.  For  the  ordinates  between  the  diverging  lines  we 
have 

P  N  _  ordinate  through  A 

C  Q  abscissa  of  A 


[H4.]  SPECIAL    CASES.  205 

But  the  part  of  the  ordinate  below  A  will  be  less  than 
that  represented  by  the  numerator  of  the  second  member 
of  this  equation,  thus  making  the  ratio  less. 

114.  Special  cases. 

1st.  Let  the  expansion  be  isothermal  ;  then  will 


(192) 

as  given  in  equation  (165)  ;  and 

Z=2*r.  (193) 

From  these  equations  the  following  table  has  been  com- 
puted, which  is  applicable  to  perfect  gases,  and  superheated 
steam  working  expansively  at  constant  temperature. 

t'  * 

Eq.  (192).  Eq.  (193) 

1,  1.000  1.00 

li  .978  1.22 

H  -937  1.41 

If  .891  1.56 

2  .846  1.69 
2i  .766  1.92 

3  .700  2.10 
3f  .644  2.25 

4  .596  2.39 
4i  .556  2.50 

5  .522  2.61 
5£  .492  2.70 

6  .465  2.79 
8  .385  3.08 

10  .330  3.30 

20  .200  4.00 

With  these  values  of  r  and  Z  the  locus   A  B  may  be 


206  HEAT  ENGINES.  [114.] 

constructed,  and  the  value  of  r  found  therefrom  will  give 
the  required  maximum  for  sensibly  perfect  gases. 

2d.  Let  the  expansion  be  adiabatic,  according  to  the 
approximate  law. 

Then 


and 

Z  =  10  -  9  /•"*» 
from  which  the  following  table  is  computed. 

£=.  Z. 

p* 

1  1.000  1.00 
li  .976  1.22 
\\  .931  1.35 
If  .844  1.54 

2  .834  1.66 
2£  .784  1.88 

3  .678  2.03 
3i  .620  2.17 

4  .571  2.29 
4£  .530  2.39 

5  .495  2.47 
5£  .464  2.55 

6  .438  2.63 
8  .357  2.86 

10  .303  3.03 

20  .177  3.55 

Results  found  by  constructing  the  locus  A  B  from  this 
table  will  be  applicable  to  saturated  steam  expanding  adia- 
batically.  Other  hypotheses  might  be  assumed,  and  corre- 
sponding results  obtained,  but  as  extremely  accurate  results 
will  not  be  expected  in  practice,  the  hypotheses  of  adia- 
batic  expansion  will  answer  for  ordinary  cases. 


[114.]  SPECIAL   CASES.  207 

In  regard  to  the  Efficiency  of  Fluid,  we  have  in  equation 
(191)  k  and  h  both  zero,  rendering  that  term  indeterminate, 
and  equation  (190)  reduces  to  infinity,  as  it  should,  since 
the  cost  of  the  steam  is  not  included  in  the  latter  prob- 
lem, which  condition  only  requires  the  most  work  per 
pound  of  steam  entering  the  cylinder.  This  requires  that 

U=(pm-F}rVl  (194) 

shall  be  a  maximum,  and  this  is  equivalent  to  making  the 
numerator  of  the  left  member  of  (190)  a  maximum,  and 
this  is  a  maximum  when  the  ordinate  P  JVis  a  maximum, 
giving  O  Q  for  the  corresponding  value  of  r.  Equation 
(194)  is  easily  reduced  for  a  maximum  for  the  isothermal 
expansion  of  gases.  For  we  have  from  equation  (164) 

u=    1  **-• 


=  O. 
which  will  be  a  maximum,  when 


(195) 

that  is,  the  ratio  of  expansion  must  be  such  as  to  reduce 
the  terminal  pressure  to  that  of  the  hack  resistance. 

If  frictional  resistances  be  neglected,  J^will  represent  the 
back  pressure,  which  will  be  that  of  the  exhaust  steam  ;  in 
which  case  we  have 


or  the  work  on  the  lack  stroke  witt  equal  the  work  done 
before  cut-off '/  hence,  the  useful  work  witt  equal  the  latent 
heat  of  expansion. 

EXERCISE. 

(The  following  exercise  is  an  abstract  of  a  paper  by 
Messrs.  Wolff  and  Denton,  Transaction  of  the  American  So- 
ciety of  Mechanical  Engineers  (1881),  147,  281,  except  that 


208  HEAT   ENGINES.  [114.] 

we  assume  that  the  engine  was  used  10  -out  of  24  hours, 
while  they  assumed  that  it  was  run  continuously.  It  was 
an  example  of  a  Buckeye  non-condensing  engine.) 

i  26"  diameter,  I  75.3  Ibs.  gauge, 

1.  Cylinders  <  ^  ^.  '     pressure  j  ^    u 


90.0    "  absolute. 

10  revolutions  per  minute,  working  10  hours  daily. 
Assume  clearance*  c  =  2£  per  cent, 

Condensation  =  30  per  cent  above  that  indicated, 
Back  pressure  =  15.7  Ibs. 
Friction  =  :     2.0   " 


.-.F  —  17.7  Ibs. 
Coal,  5  dollars  per  2000  Ibs. 
Evaporation,  9  Ibs.  of  water  per  pound  of  coal. 
We  have — 

CHARGEABLE  TO  THE  B_ILER  : 

If  the  engine  work  full  stroke,  and  no  allowance  be 
oiade  for  clearance  and  condensation,  the  cost  of  the  coal 
per  hour  will  be 

Vol.  1  stroke  feet.  Double  stroke.          Wt.  1  cu.  ft. 

~T728  ~   X  2  X  600  x  0.21185  X 

Lbs.  of  coal  per  Ib.  Cost  of  coal  dol-  Dollars  per 

of  water.  lare  per  Ib.  hour. 

i             X           -gjw  =  10.415 

Add  2^  per  cent  for  clearance .260 

Sum  =  10.675 

Add  30  per  cent  of  $10.675  for  condensation =    3.202 

Add  wages  of  fireman  ($2.25),  laborer  ($1.25) . .     =    0.146 
Interest  on  cost  of  boilers,  8500  X  0.06  -4-  (365  X  24)  =    0.058 

1             8500 
Depreciation,  say  ^  X  365  x  24 =    0.082 

Repairs,  if  $190  per  year =    0.022 

Sum  =  14.185 


L114.]  SPECIAL    CASES.  209 

CHARGED  TO  THE  ENGINE. 

Assume  cost  of  engine  when  set  to  be  $9000. 

Interest,  $9000  X  0.00  -=-  (365  X  24) =   0.063 

Wages  of  engine  runner =   0.200 

Depreciation,    say   ^-of  9000  -^  (365  X  24)  . .    ==    0.042 

Repairs,  say  $150  per  year =    0.017 

Oil  and  waste,  say =   0.030 

Sum  =  0.352. 
Then, 

F      17.7         1 


14.185  X  1728 

h_       In  __  0.7854  X  (26)'2  X  48  X  2  X  600  X  0-21185       4  X  2  X  60t 
A        v  0.352  X  144  4.7 

0".  7854  X  (26)2 

=  40.47  =  0  C. 


pl          5.09 

In  Fig.  50,  lay  off  0  C  =  40.47  and  C  D  =  7.95,  and 
from  the  point  D  thus  found  draw  the  tangent  D  P,  and 
from  P  let  fall  the  perpendicular  P  Q,  then  will  0  Q  be, 
the  ratio  of  expansion  for  this  case,  which,  for  adiabatic 
expansion,  will  be 

0  Q  =  3.4. 

In  a  class  of  forty  students  working  independently  and 
with  different  scales,  the  results  differed  only  by  two  or 
three  tenths,  using  the  same  law  of  expansion,  a  result  near 
enough  for  practice  in  the  present  state  of  the  science.  The 
exact  value  may  be  found  by  trial  and  error,  by  substituting 
in  expression  (191).  (Addenda.) 

(Literature.  General  solution  by  the  late  Professor  Rankine,  Phil. 
Mag.  (1854X  21,  176  ;  Trans.  Roy.  Soc.  Edinburgh,  Vol.  XX.,  Part  II.; 
Ship  Building ;  Miscellaneous  Scientific  Papers,  pp.  288-299.  The  En- 
gineer, 1866,  April  2d,  p.  248,  gives  a  modification  of  his  graphical 


210  HEAT    ENGINES.  [U5.J 

method.  Wolff  and  Denton,  Transaction  of  the  American  Society  of 
Meeltanical  Engineers  (1881),  pp.  147-381  (and  discussion  by  others  in  the 
same  Transactions);  Trans.  lust.  Civ.  Eng.,  London,  1881-82,  Vol.  IX., 
Part  II.,  75,  Part  III.,  44;  American  Engineer,  1881,  June,  July,  Aug., 
Xov.  Emery,  Jour.  Frank.  last.,  May,  1875.  Marks,  ibid.,  1880,  '83, 
'84.  Thurston,  ibid.,  1880,  '81,  '83,  '84.  Wood,  ibid.,  May,  1884.) 

2.  Let  the  dimensions  of  the  engines  and  cost  of  plant  be 
as  in  the  preceding  exercise,  the  gauge  pressure  70.3  Ibs., 
10  revolutions  per  minute,  working  2±  hours  daily,  conden- 
sation 25  per  cent,  back  pressure  F.  =   18  Ibs.,  coal,  evap- 
oration, oil,  waste,  and  interest  as  in  the  preceding  exercise ; 
also  wages,  except  that  they  are  for  12  hours  instead  of  10  ; 
life  of  boiler  10  years,  repairs  $225 ;  life  of  engine  20  years, 
repairs  of  engine  $175  per  year  ;  engine  to  run  300  days  of 
the  year  ;  find  the  most  economical  point  of  cut-off. 

3.  If  the  cost  of  producing  the  steam  be  neglected,  or  h 
=  0,  in  Exercise  1,  find  the  proper  point  of  cut-off. 

4.  In  Exercise  1  if  the  "  cost  of  the  engine  "  be  neg- 
lected, or  k  =  0,  find  the  proper  value  for  r.     (Use  the  left 
member  of  (190).) 

115.  Multiple  expansions. — Engines  are  made 
with  two  or  more  cylinders,  so  arranged  that  after  steam  has 
done  some  work  in  one  cylinder  it  may  be  exhausted  into 
another,  and  from  the  second  into  the  third,  and  so  on,  and 
the  expansions  continued  in  the  successive  cylinders.  If 
two  cylinders  are  employed,  the  combination  is  called  a  com- 
pound engine  /  if  three  cylinders,  triple  expansion  /  if  four, 
quadruple  expansion.  Since  multiple  expansions  have  come 
into  practice,  it  might  be  well  to  drop  the  term  compound, 
and  substitute  double  expansion.  The  cylinders  may  be 
arranged  in  any  desirable  manner.  If  placed  end  to  end, 
having  a  common  piston  rod,  they  are  called  tandem.  They 
may  be  placed  side  by  side,  close  to  each  other,  or  separated 
many  feet  and  connected  by  a  large  pipe.  The  cylinder 
first  receiving  steam  is  called  the  high  pressure  cylinder,  and 


[115.]  MULTIPLE   EXPANSIONS.  2)1 

has  the  smallest  piston  displacement  per  stroke,  and  the 
others  the  low  pressure  cylinders,  each  increasing  in  size  as 
they  are  more  remote  in  the  grade  of  expansion  from  the 
high  pressure.  In  triple  expansions,  the  smaller  cylinder  is 
called  the  high  pressure,  the  next,  the  intermediate,  and  the 
third  or  largest,  the  low  pressure  cylinder.  The  high-press- 
ure cylinder  may  exhaust  directly  into  the  next  one,  or  into 
a  receiver,  and  in  the  same  manner  from  the  next  cylinder, 
and  so  on.  In  some  cases  double  expansion  is  accomplished 
in  three  cylinders,  the  high-pressure  cylinder  being  between 
the  two  low-pressure  cylinders. 

If  the  fluid  retained  its  state  of  aggregation,  there  would 
be  no  theoretical  gain  in  expanding  in  two  or  more  cylin- 
ders over  that  of  expanding  in  one  between  the  same  limits 
of  temperature ;  but,  on  the  other  hand,  there  would  be  a 
loss,  for  the  spaces  between  the  cylinders  serve  as  clearances 
which  must  be  filled  with  steam.  Steam  cards  from  multi- 
ple-expansion engines  clearly  show  this  loss  ;  yet  experi- 
ence proves  that  there  is  a  gain  of  efficiency.  This  is  chiefly 
due  to  the  fact  that  liquefaction  of  the  steam  is  less  when 
expanded  in  several  cylinders,  for  the  walls  of  the  cylinders 
are  kept  at  a  more  nearly  uniform  temperature,  being  more 
nearly  that  at  which  the  steam  enters  the  cylinder. 

There  is  also  a  mechanical  advantage,  since  the  initial 
stress  on  the  crank  pins  will  not  be  so  excessive.  With 
triple  expansions,  the  initial  stress  on  the  crank  pin  may  be 
about  one-half  or  one-third  of  what  it  would  be  if  expansion 
were  made  in  one  cylinder  only. 

This  arrangement  also  produces  a  more  uniform  rotation 
of  the  shaft,  which  in  the  case  of  vessels  driven  by  propel- 
lers is  favorable  to  greater  efficiency  of  speed.  So  that  if 
a  single  expansion  and  a  triple  expansion  should  show  the 
same  economy  of  fuel  per  horse-power,  the  triple  expansion 
in  the  same  vessel  ought  to  show  greater  economy  of  fuel 
for  a  given  mileage. 


212  HEAT   ENGINES.  [116.] 

116.  Condensation. — The  laws  which  govern  the 
liquefaction  of  steam  in  the  cylinder  are  not  well  known. 
Theory  recognizes  three  sources  for  the  appearance  of  water 
in  the  cylinder :  Jirst,  water  carried  from  the  boiler  to  the 
cylinder  in  the  form  of  a  spray,  in  which  particles  of  liquid 
water  are  mingled  with  the  steam ;  second,  liquefaction  pro- 
duced by  the  expansion  of  saturated  steam ;  and,  third, 
liquefaction  produced  by  the  walls  of  the  cylinder.  The 
first  pertains  chiefly  to  the  construction  and  management  of 
the  boiler ;  the  second  has  been  discussed  from  a  theoretica" 
standpoint  by  Rankine  and  Clausius,  as  stated  in  Article  98 
Rankine,  in  an  example  with  assumed  data,  in  which  tin* 
ratio  of  expansion  was  32^,  found  that  nearly  18  per  cent 
of  the  steam  entering  the  cylinder  was  liquefied  during  ex 
pansioH  from  this  caiise  (Misc.  Sc.  Papers,  p.  399).  Theon 
shows  that  superheated  steam  loses  nothing  from  this  caust> 
so  long  as  it  remains  above  the  condition  of  saturation,  and 
actual  engines  confirm  this  result.  Calorimeter  tests  oi 
steam-jacketed  engines  have  shown  a  total  loss  from  lique 
faction  of  from  10  to  20  per  cent. 

The  third  has  been  discussed  by  Professor  Cotterell  in  his 
work  on  The  Steam-Engine, '  pp.  246-269,  in  which  he 
shows  that  this  may  be  the  principal  cause  of  the  loss  of 
efficiency  of  the  fluid.  The  l:ws  of  conduction  and  radi- 
ation are  not  sufficiently  well  known  to  enable  one  to  estab- 
lish a  complete  theory  of  liquefaction  in  this  regard  ;  and 
if  they  were,  the  variations  of  temperature  due  to  expansion, 
as  well  as  the  varying  temperature  of  the  walls  from  the  be- 
ginning to  the  end  of  the  stroke,  would  greatly  complicate 
the  problem. 

If  the  liquefied  water  be  deposited  upon  the  inner  sur- 
face of  the  cylinder,  as  it  will  be  in  the  third  case,  it  will 
facilitate  the  conduction  of  heat,  and  the  result  will  be  very 
different  from  the  condition  in  which  the  water  remains 


[117.]  EXPLOSIVE   GAS   ENGINE.  213 

distributed  throughout  the  steam,  as  it  is  supposed  to  be  in 
the  first  and  second  cases  named  above. 

The  reduction  of  temperature,  due  to  the  action  of  the 
walls,  would  also  have  an  influence  upon  the  theory  in- 
volved in  the  second  case,  in  a  manner  which  has  not  yet 
been  considered. 

Initial  condensation  is  that  which  takes  place  during  the 
admission  of  steam,  and  is  due  chiefly  to  exposure  to  sur- 
faces colder  than  the  steam,  and  is-  independent  of  the 
case  investigated  by  Rankine.  It  can  be  reduced  by  keep- 
ing the  walls  at  nearly  the  temperature  of  the  entering 
steam,  and  hence  may  be  nearly  prevented  by  a  steam-jacket, 
and  in  other  cases  may  be  reduced  by  late  cut-off  arid  high 
speed.  The  economy  of  high  expansion  is  so  well  es- 
tablished by  theory  and  confirmed  by  experience,  when 
condensation  is  avoided,  that  other  means  than  that  of  a  late 
cut-off  will  be  sought  for  preventing  liquefaction. 

Theory  does  not  enable  us  to  compute  the  amount  of 
condensation  for  any  particular  case ;  it  must,  therefore,  be 
determined  by  direct  experiment.  A  few  examples  are  given 
Jn  the  following  notes. 

NOTES. 

117.  Experiments  on  steam-engines: 

(a.)  Hirn's  experiments. — By  far  the  most  complete  set  of 
experiments  scientifically  conduced  were  those  under  the 
direction  of  M.  Hirn,  by  MM.  O.  Hallauer,  "W.  Grosse- 
teste,  and  Dwelshauverse  Dery,  begun  in  1873,  and  extending 
over  several  years.  The  results  of  the  experiments  are  pub- 
lished in  the  Bulletin  Special  of  the  Societe  IndustrieUe  de 
Mulhouse,  1876.  Smith  on  Steam  Using  contains  a  sum- 
mary of  these  experiments,  pp.  188-285. 

(5.)  Navy  experiments. — Mr.  B.  F.  Isherwood,  while  chief 


214  HEAT   ENGINES.  [117.] 

engineer  of  the  United  States  Navy,  made  extensive  experi- 
ments upon  the  boilers  and  engines  of  several  steamships 
of  the  navy,  under  the  general  direction  of  the  Department 
of  the  Navy,  which  were  published  in  two  large  volumes, 
entitled  Experimental  Researches  in  Steam  Engineering, 
Philadelphia,  1863,  1865. 

Mr.  Isherwood  made  the  first  attempt,  so  far  as  known, 
to  determine  the  amount  of  liquefaction  of  steam  in  un- 
jacketed  cylinders  for  various  ratios  of  expansion.  The 
experiments  were  made  on  the  engines  of  the  steamer 
Michigan,  in  1861,  and  showed  a  large  amount  of  liquefac- 
tion, increasing  at  low  speeds  and  high  expansions.  These 
were  discussed  by  Rankine,  and  published  in  the  Proceed- 
ings of  the  Institution  of  Engineers,  Scotland,  1861-62. 

(f.)  Navy  experiments — continued. — Messrs.  Emery  and 
Loring,  in  1874,  experimented  on  the  engines  of  the  United 
States  revenue  steamers  Hache,  Rmh,  Dexter,  and  Galla- 
tin,  the  reports  for  which  were  published  by  the  Govern- 
ment, and  also  in  Engineering,  Yols.  XIX.  and  XXI. 

From  these  and  other  experiments,  some  have  concluded 
that,  for  unjacketed  cylinders,  it  will  be  sufficient  to  allow 
for  the  liquefaction  of  steam  in  the  cylinder,  due  to  all 


For  full  stroke,     12  per  cent  of  the  total  feed  water, 

2  expansions,  23 

4        '  36 

6        '  54 

8        '  67 

10        •  72 

12        '  75 

(American  Engineer,  1884,  May  23,  p.  207.) 

For  dry  steam  at  high  temperatures  these  allowances  are 
probably  much  too  large,  although  they  may  sometimes  be 
realized  with  saturated  vapor. 

(d}  Experiments  at  the  Stevens  Institute.     Messrs.  Gately 


[117.]  EXPERIMENTS    ON    STEAM-ENGINES.  215 

and  Kletzscli  experimented  upon  a  Harris-Corliss  engine  hav- 
ing an  18-incli  cylinder,  42 -inch  stroke,  for  the  purpose  of 
determining  the  laws  of  condensation  under  different  condi- 
tions. The  engine  was  not  jacketed,  but  was  covered  with 
lagging  and  a  non-conducting  substance.  They  found  that 
if 

y  =  cut-off  =  0.13,  then,  cylinder  condensation  =  x  =  0.50 
"      =  0.225     "  "  "  =  0.41 

"      =  0.33      "  "  "  "  =  0.34 

"      =  0.45      "  "  "  =  0.27 

"      =  0.59      "  "  "  =  0.225 

which  values  are  well  represented  by  the  equation 
(a?  +  0.12)  (y  +  0.44)  =  0.3543. 

If  z  =  the  area  of  the  surface  exposed  in  square  feet  and 
i«  the  per  cent  of  condensation,  as  before,  the  experiments 
gave 

a?  s  —  1.026  x  —  4.Y7  z  =  221.36. 

Yarying  boiler  pressures  gave 

p  =  pressure  —  80.00  pounds,  x  =  per  cent  cjl.  condensation  =  35.24 


=  66.85 
=  52.33 
=  37.00 
=  22.30 


=  37.83 
=  36.84 
=  41.43 
=  41.19 


which  may  be  represented  by  the  equation 
x  —  45  —  0.1266  p. 

(Graduation  Thesis,  1884  ;  Jour.  Frank.  Iwt.,  1885,  Oct.,  Nov.  and 
Dec.) 

Messrs.  Blauvelt  and  Haynes,  by  calorimeter  tests  upon 
the  engines  of  the  steamship  Hudson  of  the  Cromwell  Line, 
found  in  some  cases  only  10  per  cent  of  liquefaction  when 
the  cut-oil  was  -j^.  The  pistons  were  48  inches  in  diam- 
eter, stroke  6  feet;  steam-jacketed  cylinder;  800  horse- 
power engine.  (Thesis,  1886.) 

The  experiments  of  Mr.  James  S.  Merritt  upon  a  small 
direct-acting  steam  pump,  at  various  piston  speeds,  the  di- 


216 


HEAT    ENGINES. 


[118.] 


ameter  of  the  steam  cylinder  being  eight  inches,  and  the 
stroke  about  9J  inches,  gave  the  following  results  : 


Water  consumed 

Double  strokes 
per  minute. 

Indicated  horse- 
power. 

per  I.  H.  P.  per  hour. 

Steam  condensed 
in  cylinder  ;  per  cent 
of  the  feed  water. 

Calculated 

Actual 

pounds. 

pounds. 

10 

1.25 

46 

114 

60 

20 

2.36 

50 

89 

43 

80 

3.63 

48 

100 

52 

40 

4.92 

48 

75 

36 

50 

5.34 

51 

86 

40 

(Thesis,  ISS6.). 

These  results  are  not  as  uniform  as  is  desirable  in  such  an 
experiment.  The  one  for  20  strokes  appears  to  be  excep- 
tional. 

Messrs.  McElroy  and  Parsons,  in  their  test  of  the  boilers 
ind  compound  engines  of  the  tug  Blue  Bonnet  found 
that  37  per  cent  of  the  feed  water  was  unaccounted  for  by 
the  indicator  cards.  The  cylinders  were,  respectively,  19" 
X  22"  and  22"  X  22".  The  cards  indicated  that  twenty 
pounds  of  water  per  horse-power  per  hour  were  consumed. 
Boiler  pressure,  70  Ibs.  (Thesis,  1887.) 

(e)  Cylinder  condensation.  A  series  of  experiments  to 
determine  the  amount  of  cylinder  condensation  were  made 
by  Major  English,  England,  and  published  in  Engineering, 
1887.  It  is,  however,  questionable  whether  they  are  of 
much  practical  value,  since  the  conditions  do  not  conform  to 
actual  practice. 

118.  Miscellaneous: 

(/")  The  weight  of  steam  required  by  an  engine  is  in- 
dicative of  its  efficiency.  Eighteen  pounds  per  horse-power 
per  hour  is  very  good  practice.  Thirty-five  to  fifty  pounds 
is  common  in  ordinary  practice. 

(g)  At  Calumet,  Mich.,  an  engine  was  tested  working  at 


LI  18.] 


MISCELLANEOUS. 


217 


about  600  horse-power,  for  over  ten  days,  which  ran  with 
16.3  pounds  of  feed  water.  (Am.  Soc.  Mechanical  Engi- 
neers, Discussion,  Hartford  meeting,  p.  14.) 

(A)  In  a  three  days'  test  of  the  steamship  Para,  having 
triple-expansion  engines,  the  actual  weight  of  steam  con- 
sumed per  I.  II.  P.  per  hour  was  13.4  Ibs.  Adding  15  per 
cent  for  initial  condensation  gives  15.4  Ibs.;  coal,  1.54  Ibs. 
per  indicated  horse-power  ;  evaporative  power  of  the  coal, 
12.0  Ibs.  from  and  at  212°  F.  The  Stella  consumed  13.7 
Ibs.  of  steam  per  I.  II.  P.,  with  1.36  Ibs.  of  coal  whose 
evaporative  power  was  13.9  Ibs.  of  water  from  and  at  212° 
F.  (Proc.  Institution  Mech.  Eng.,  1886-87,  pp.  492-506.) 

(i)  Large  Ocean  Steamers. 


City  of  Rome. 

Umbria  and 
Mruria. 

Servia. 

Length  feet     

542.5 

500 

515 

Breadth   "         

52.0 

57 

52 

11230 

9860 

10  960 

Indicated  H   P 

11890 

14321 

10  300 

Speed   miles  per  hour  

18.2 

20  2 

17 

Coal   per  day  tons    

185 

315 

205 

Coal  per  I   H.  P      

2.2 

2  1 

2 

n  i  •   j        )  Diana,  ins  

(3@46 

1  @     71 

1  @     72 

Cylinders  \ 
(  Stroke   " 

(3  @  86 

72 

2  @  105 

72 

2  @  100 

78 

Steam  pressure  Ibs 

90 

110 

(j }  In  the  year  1840,  the  time  of  crossing  the  Atlantic 
Ocean  in  a  steamship  was  about  13  days. 
The  recent  short  passages  have  been : 

City  of  Rome 6  d.          18  h.  Om. 

Oregon 6  d.          10  h.          35  m. 

Etruria 6  d.  5  h.          31  m. 

(Scribner's  Magazine,  1887,  p.  315.) 

In  1887,  beginning  May  28th,  the  time  of  the  Umbria 
was  6  d.,  4  h.,  12  m.  In  1888,  beginning  May  24th  at 
Queenstown,  the  time  of  the  Etruria  was  6  d.,  1  h.,  55  m. 


218  HEAT    ENGINES.  [118.] 

Distance,  3028  miles ;  average  speed,  20.7  miles  per  hour. 
On  June  1st  the  average  speed  was  24.08  miles  per  hour. 

(K)  Count  de  Pambour  represented  that  the  friction  of  an 
engine  increased  with  the  load,  the  expression  for  the  resist- 
ance being  of  the  form 

R  =  J?0  +  x  B, 

in  which  x  is  a  coefficient,  greater  than  unity,  R  the  varia- 
ble load,  and  7?0  the  resistance  of  the  unloaded  engine. 

But  Mr.  Charles  T.  Porter,  about  1871,  wrote  that  "  ex- 
periments with  a  friction  brake  have  shown  no  appreciable 
difference  between  the  losses  of  power  in  friction  when  very 
small  and  very  large  loads  were  driven  by  the  same  engine. 
(American  Machinist,  Dec.  25,  1886,  p.  8.) 
•  The  result  of  Porter's  experiments  has  more  recently 
been  confirmed  by  several  other  experimenters.  (Trans. 
Am.  Sac.  Mech.  Eng.,  Vol.  VII.,  pp.  86-113.) 

(7)  The  Stiletto  is  a  torpedo  boat  95  feet  long,  weighs 
28  tons,  develops  450  H.  P.,  having  16  H.  P.  per  ton  of  dis- 
placement, works  with  150  pounds  pressure,  ran  30  miles  in 
77  minutes,  or  an  average  of  23.7  miles  per  hour,  passing 
the  Mary  Powell,  hitherto  the  fastest  boat  on  the  Xorth 
River.  The  highest  speed  attained  is  not  given.  (The 
Mech,  Eng.,  June  27,  1885.) 

(m)  The  compound  locomotives  tried  on  the  Boston  and 
Albany  Railroad  failed  as  economizers  of  fuel,  and  were 
changed  to  the  ordinary  form. 

(n)  Steam  pressure  used  in  marine  engines  has  gradually 
increased  at  an  average  rate  of  more  than  two  pounds  per 
year  for  many  years.  The  following  values  are  given  in 
Scribner's  Magazine  for  May,  1887. 


[118.]  MISCELLANEOUS.  219 


1825  

2  to  3 

1830  

5 

1835  

.  8 

1840  

10 

1845  

14 

1850  

21 

1855  

.  25 

1860  

30 

1865  

40 

1870  

50 

1875  

60 

1880  

70 

1882  

80 

1886  

150  to  160 

The  writer  does  not  clearly  state  why  these  particular 
values  are  given.  Some,  though  relatively  few,  steamships 
carry  160  pounds  pressure,  and  if  this  be  the  highest,  then, 
on  the  same  plan,  the  highest  for  preceding  years  ought  to 
have  been  given.  Many  locomotive  boilers  carry  160  pounds 
pressure,  and  have  done  so  for  several  years. 

(0)  Ideal  efficiency.  It  is  sometimes  convenient  for  refer- 
ence to  have  an  ideal  maximum  efficiency  for  steam  powers. 
Assume,  then,  that  the  steam  pressure  is  200  pounds  per 
square  inch  by  the  gauge,  and  that  the  back  pressure  is 
one  pound  per  square  inch  absolute,  and  that  the  forward 
pressure  decreases  to  one  pound  absolute  ;  then  will  the 
temperature  corresponding  to  the  higher  pressure  be  388° 
F.  and  to  the  lower  102°  F. ;  and  the  maximum  efficiency 
when:  working  between  these  temperatures  will  be 

388  —  102  _  286 

388  +  460  =  848  =  °'325' 

which  is  somewhat  less  than  \.  To  realize  this  result 
would,  according  to  the  approximate  adiabatic  law,  require 
about  118  expansions,  which  fact  alone  shows  that  such  an 
efficiency  is  far  beyond  existing  possibilities  in  a  working 
engine.  The  waste  of  heat  in  the  furnace  of,  say,  25  per 
cent,  loss  of  pressure  between  the  boiler  and  the  engine, 
loss  of  heat  at  the  exhaust,  and  other  losses  reduce  the 


220  HEAT    ENGINES.  [11H.] 

efficiency  of  steam  plants  below  15  per  cent  of  the  heat 
energy  of  the  fuel. 

(p)  Quadruple-expansion  engines.  In  the  American 
Machinist  of  December  3d,  1887,  is  an  article  taken  from 
London  Engineering,  describing  a  system  of  quadruple- 
expansion  marine  engines,  which  have  been  placed  on  sev- 
eral new  steamers.  The  boilers  for  these  engines  are  de- 
signed to  carry  180  Ibs.  pressure.  It  is  claimed  that  these 
are  6  to  8  per  cent  more  efficient  than  triple-expansion  en- 
gines. 

(q)  Efficiency  of  plant.  Take  the  case  of  the  steamship 
Ohio,  of  the  International  Steamship  Company,  which  has 
triple  expansion  engines  of  2100  I.  H.  P.,  the  gross  tonnage 
of  the  vessel  being  3325  tons.  The  engines  were  guaranteed 
to  consume  not  more  than  1.25  Ibs.  of  coal  per  I.  H.  P.  per 
hour.  The  cylinders  were  31  in.,  46  in.,  72  in.,  and  51  in. 
stroke. 

The  trial  trip  developed  an  I.  H.  P.  for  1.23  Ibs.  of  coal.* 

The  caloritic  capacity  of  this  coal  is  not  known  to  us,  but  it  is 
quite  certain  that  on  such  a  trial  the  best  of  coal  would  be  used. 

If  the  heat  of  combustion  was  15000  thermal  units — 
which  is  a  very  high  value — there  would  have  been  ex- 
pended 

1.23  X  15000  =  18450  thermal  units 
per  horse-power  per  hour,  or 

18450  X  778  =  14354100  foot-pounds 
of  energy  to  produce  one  horse-power,  or 

33000  X  60  =  1980000  foot-pounds 
of  work  per  hour,  in  which  case  the 

M  •  si  1980000        A1oft 

ejficiency  of  plant  =  1435410Q  =  °-138> 

or  nearly  14  per  cent. 

*  We  Mechanical  Engineer,  Sept.  10th,  1887,  pp.  49.  50,  taken  from 
Jndustrtot 


[118.J  MISCELLANEOUS.  221 

If  the  coal  contained  14000  thermal  units — a  fair  value 
for  very  good  coal — the 

„,  .  „    7  1980000 

efficiency  of  plant  =  133{moo  =  0.148, 

or  nearly  15  per  cent. 

The  boilers  would  naturally  be  in  excellent  condition  for 
such  a  trial,  and  if  they,  including  the  steam  connections  up 
to  the  steam-chest,  gave  an  efficiency  of  75  per  cent,  then 
we  would  have  for 

of  the  engines  only,  •  '  '     =  0.184  in  former  case 

75 

=  0.197  "   latter     " 


75 

If  the  efficiency  of  the  boiler  and  connections  were  70 
per  cent — a  fair  value — the  efficiency  of  the  engines  would 
be  0.197  in  the  former  case  and  0.211  in  the  latter. 

It  is  a  remarkably  good  plant  that  will  produce  an  indi 
cated  horse-power  with  1.23  pounds  of  the  best  coal. 
While  it  is  well  known  that  such  a  trial  may  be  so  con- 
ducted as  to  give  a  result  too  favorable  to  the  contractors — 
by  not  giving  proper  credit  to  the  heat  generated  just 
before  starting,  or  by  letting  the  fires  run  too  low  at  the 
close,  or  by  not  standardizing  the  indicator,  &c. — yet,  on 
the  other  hand,  the  proprietors  of  the  vessel  would  naturally 
check  all  the  conditions  so  as  to  determine  for  themselves 
if  the  terms  of  the  contract  were  fulfilled.  We  therefore 
feel  some  confidence  that  marine  steam  plants  have  been 
made  that  have  developed  an  actual  efficiency  of  some  14 
or  15  per  cent ;  and  certainly  the  conventional  10  per  cent 
efficiency  used  by  popular  writers  is  exceeded  in  some 
cases. 

Tests  of  commercial  coal  taken  at  random  show  that  the 
heat  of  combustion  frequently  falls  below  12000  thermal 
units  per  pound.  Ocean  steamships  have  been  reported  as 


222  HEAT   ENGINES.  [118.] 

developing  an  I.  II.  P.  per  hour  with  2.1  pounds  of  coal, 
and  in  some  cases  with  less  than  that  amount  ;  hence  if  this 
coal  contained  12000  thermal  units,  the  efficiency  would  be 

19800000 
196000000  ~ 

or  over  10  per  cent. 

If  the  initial  pressure  in  the  cylinder  were  160  pounds 
to  the  square  inch  absolute  (about  145  gauge  pressure)  and 
back  pressure  3  pounds,  and  the  steam  saturated,  then 
would  the  initial  temperature  be  363°  F.  and  the  lower 
142°  F.,  and  if  this  heat  were  used  in  a  perfect  elementary 
engine  between  these  limits,  the  efficiency  would  be 


and  TO  per  cent  of  this  is  0.1876,  which  is  nearly  one  of  the 
values  found  above. 

(r)  The  torpedo  boat  Ariete,  built  for  the  Spanish  Gov- 
ernment, has  twin  screws  and  compound  engines,  14f",  24^", 
by  15"  stroke.  Average  boiler  pressure,  152  Ibs.,  revolu- 
lutions,  395  per  minute,  developing  155  II.  P.,  and  steam- 
ing 24.9  knots  (28.8  miles)  for  two  hours  continuously. 
One  measured  mile  was  run  at  the  rate  of  26  knots  (30.1 
miles)  per  hour. 

(s)  Steamer  Anthracite,  in  1880,  had  triple  expansion  en- 
gines. 
Pounds  of  water  per  I.  H.  P.  per  hour  ............      21.68 

Evaporation  per  Ib.  of  coal  at  and  from  212°  ......       9.27 

u     "    "  combustible     u       "     ......     11.25 

Steam-pressure  in  the  boiler,  gauge,  Ibs  ............   216.5 

"  "    "    1st  cylinder,  gauge,  Ibs  ......   201.6 

Terminal  pressure  in  the  3d  cylinder,  gauge,  Ibs.  .  .  .       9.55 

Expansion,  times  ..............................     25.7 

Coal  per  I.  H.  P.  per  hour,  Ibs  ...................    *  2.61 


[119.]  STIRLING'S  HOT-AIR  ENGINE.  223 


HOT-AIR  ENGINES. 

119.  Stirling's  (or  Lauberean's)  Hot-air  En- 
gine. This  engine  was  invented  by  Dr.  Robert  Stirling 
about  the  year  1816,  and  improved  by  his  son,  Mr.  James 
Stirling ;  for  the  details  of  which  see  Proceedings  of  tlie 
Institution  of  Civil  Engineers,  1845.  It  was  further  im- 
proved by  M.  Laubereau,  the  form  of 
which  is  shown  in  Fig.  51.  It  consists 
of  two  cylinders  of  different  diame- 
ters, having  a  free  communication  be- 
tween them.  The  smaller  piston,  B, 
Fig.  52,  is  the  working  piston,  and 
drives  the  engine.  The  larger  piston 
or  plunger,  A,  is  made  chiefly  of  plaster 
of  Paris  or  other  non-conductor  of  heat, 
and  is  somewhat  smaller  than  the  bore 
of  the  cylinder,  so  that  the  air  may  pass  FIG.  51. 

freely  past  it.     Also  an  annular  space 

about  the  cylinder  is  filled  with  thin  plates  or  small  wires 
which  heat  quickly  as  the  hot  air  passes  among  them, 
and  as  quickly  give  up  their  heat  to  the  cold  air  on  its 
return.  This  device  is  the  regenerator  referred  to  in 
Article  105.  The  top  of  the  cylinder  at  C  may  be  made 
double  to  admit  of  the  passage  of  water  ;  or,  what  is  bet- 
ter, the  upper  end  of  the  cylinder  may  be  filled  with  an 
extensive  coil  of  small  copper  tubes  through  which  water 
is  made  to  flow  by  means  of  a  force-pump  worked  by  the 
engine,  the  object  being  to  maintain  a  low  temperature 
in  that  end  of  the  •  cylinder,  and  thus  cool  the  air  at  that 
end,  and  hence  is  called  the  refrigerator.  The  object  of  the 
plunger  is  to  transfer  a  mass  "of  air  from  one  end  of  the  cyl- 
inder to  the  other  and  back  again,  and  so  on  alternately,  which 
is  accomplished  by  the  reciprocating  motion  of  the  plunger. 
The  plunger  is  sometimes  called  the  displacing  piston. 


224  HEAT   ENGINES.  [130.] 

The  plunger,  J.,  is  operated  by  a  cam  so  constructed  and 
arranged  that  when  the  piston,  £,  is  near  its  upper  dead 
point,  the  plunger  will  be  driven  very  quickly  to  the  lower 
end  of  its  stroke  and  remain  there  until  the  piston  descends 
to  near  its  lower  dead  point,  when  the  plunger  will  be 
driven  quickly  to  the  upper  end  of  its  stroke. 

The  cylinder  containing  the  plunger  is  called  tfte  receiver. 
The  mass  of  air  in  the  entire  engine  is  constant,  and  is  so 
maintained  by  a  small  air-pump  worked  by  the  engine, 
forcing  air  into  the  passage  D.  The  mass  of  air  in  the 
lower  part  of  the  receiver,  and  which  is  referred  to  as  being 
below  the  plunger,  when  the  plunger  and  piston  are  at  th<t 
upper  ends  of  their  strokes,  is  called  the  working  air,  and  al 
the  other  air  is  called  cushion  air.  These  masses  of  air  an 
not  separated,  neither  do  they  keep  entirely  separate  fron. 
each  other,  but  intermingle  somewhat.  When  both  pistons 
are  at  the  upper  ends  of  their  strokes,  the  entire  mass  of  air 
will  have  its  greatest  volume,  and  that  which  we  consider 
working  air  will  have  its  greatest  volume ;  but  when  the 
plunger  is  up  and  piston  down,  some  of  that  which  we  call 
cushion  air  will  be  forced  into  the  lower  part  of  the  re- 
ceiver, and  thus  become  virtually  working  air.  This  latter 
quantity  will  be  variable,  and  will  not  be  considered  as 
working  air. 

The  cushion  air  will  be  at  nearly  uniform  temperature, 
and  will  be  considered  the  same  as  that  of  the  refrigerator. 

To  make  this  engine  double  acting  requires  two  receivers 
having  plungers  working  in  opposite  directions,  one  receiver 
being  connected  with  the  upper  end  of  the  working  cylin- 
der and  the  other  with  the  lower. 

12O.  Theory  of  Stirling's  Engine,  or  of  a  hot- 
air  engine  in  which  changes  of  temperature 
of  the  working  fluid  are  made  at  sensibly  con- 
stant volume.  In  this  analysis  we  avoid  the  refine- 
ments which  would  result  from  following  exact  conditions 


[120.] 


THEORY    OF   STIRLING'S    ENGINE, 


225 


by  assuming  ideal  conditions,  which  will  represent  approxi- 
mately the  real  ones.  For  this  purpose  we  make  the  follow- 
ing assumptions : 

(a)  That  the  working  air  is  that  under  the  plunger  when 
both  the  piston  and  plunger  are  at  the  upper  ends  of  their 
strokes,  and  the  working  air  is  at  its  highest  temperature. 

(&)  That  the  cushion  air  remains  at  constant  temperature — 
thus  neglecting  the  fact  that  a  part  of  it  enters  the  receiver 
and  virtually  becomes  working  air. 

(<?)  That  the  mass  of  working  air  is  transferred  instantly, 
and  that  the  changes  of  its  temperature  are  also  instanta- 
neous. 

(d)  The  air  in  the  clearance  at  the  lower  end  of  the  re- 
ceiver is  discarded,  but  might  be  included  in  the  cushion  air. 

Other  assumptions  will  be  made  as  the  subject  is  de- 
veloped. 

In  Fig.  53,  let  A  represent  the  state  of  the  working 
fluid  when  at  its  least  volume  and  greatest  pressure;  in 
which  condition  the  temperature  will  be  greatest  and  the 
working  piston  will  be  at  the  bottom  of  its  stroke,  the 


F  A  PE 


A      E 


FIG.    52. 


O          S  VW  X    Y 

FIG.    53. 


FIG.    54. 


volume  in  the  receiver  and  clearances  being  It  E.  As 
the  working  piston  rises,  the  plunger  remaining  at  the 
top,  the  air  will  expand  at  a  constant  temperature  to  the 
state  J5,  the  path  of  the  fluid  being  the  isothermal  A  B. 
Now  let  the  plunger  be  suddenly  depressed — the  working 
air  will  at  once  be  transferred  to  the  upper  end  of  the  re- 


226  HEAT    ENGINES.  [120.] 

ceiver  and  there  cooled  by  the  refrigerator.  If  the  piston 
were  stationary,  the  volume  of  the  working  air  would  di- 
minish and  its  path  would  be  some  line  inclined  to  the  left 
of  B  C\  but  in  order  to  secure  the  ideal  condition  of  a- 
change  of  temperature  at  constant  volume,  conceive  that  the 
motion  of  the  piston  is  so  regulated  as  to  relieve  the  pres- 
sure on  the  working  air  in  such  a  way  as  to  make  the  path 
of  the  fluid  a  straight  line,  B  C,  perpendicular  to  the  axis 
of  volumes. 

To  accomplish  this,  the  working  piston  must  sweep  over 
the  volume  X  Y  while  the  pressure  is  falling  from  B  to  C. 
When  the  piston  descends,  sweeping  over  the  volume  Y  TF, 
the  working  fluid  will  be  compressed  at  the  constant  tempera- 
ture of  the  refrigerator  to  the  state  D,  the  path  being  the 
isothermal  C  D.  At  the  state  D,  let  the  plunger  be  sud- 
denly raised  to  the  top  of  the  receiver,  while  the  piston 
moves  according  to  such  a  law  that  the  change  of  tempera- 
ture will  be  made  at  constant  volume.  A  B  C  D  will  be 
an  ideal  diagram  of  the  working  fluid  under  the  conditions 
imposed.  The  motion  here  imposed  upon  the  piston  is  only 
a  rough  approximation  to  its  actual  motion. 

Prolong  G  B  to  P ;  then  will  R  P  be  the  greatest  vol- 
ume of  the  working  air,  which,  according  to  the  hypothesis, 
occupies  the  lower  part  of  the  receiver  when  the  piston  is 
near  the  upper  end  of  its  stroke.  Between  the  plunger, 
when  at  the  upper  end  of  its  stroke,  and  the  piston,  when  at 
its  lower  end,  will  be  the  space  at  the  upper  end  of  the 
receiver,  the  space  in  the  ports  and  passage  D,  Fig.  52,  and 
the  clearance  below  the  piston  B.  Take  P  E  to  represent 
the  volume  of  these  spaces.  The  diagram  shows  that  with 
this  arrangement  none  of  the  so-called  working  air  will 
enter  the  working  cylinder,  and  the  latter  may  be  kept  cool 
to  facilitate  lubrication.  The  volume  P  E  should,  in  prac- 
tice, be  small  compared  with  the  volume  of  the  lower  part  of 
the  receiver,  say  from  5  to  15  per  cent  of  the  volume  KB. 


[120.]  THEORY   OF   STIRLING'S   ENGINE.  227 

To  find  the  path  of  the  cushion  air,  take  It  J^  =  A  E\ 
then,  according  to  supposition  &,  the  equilateral  hyperbola 
FJ  will  be  the  isothermal  representing  the  changes  of 
pressure  and  volume  of  cushion  air.  If  the  weight  of 
cushion  air  was  the  same  as  that  of  the  working  air,  F '  J 
would  fall  upon  D  C\  but  as  it  is  generally  less,  it  is  placed 
below. 

To  construct  the  ideal  indicator  diagram,  make  AE  —  RF, 
B  N=  KG,  CQ  =  MI,  D  U  =  L  S;  then  will  E, 
N,  Q,  £7"be  corners  in  the  ideal  diagram  that  should  be  de- 
scribed by  the  working  engine  under  the  conditions  imposed. 
In  an  actual  diagram  the  corners  are  rounded. 

To  find  the  efficiency,  let 

TI  =  the  highest  absolute  temperature  of  the  working  air, 

ry  =  the  lowest         "  "  "     "         "          " 

r  =  the  ratio  of  expansion. 

Since  the  gas  is  sensibly  perfect,  and  the  expansion  along 
A  B,  Fig.  53,  is  isothermal,  we  have,  for  the  heat  absorbed 
from  the  furnace,  equation  (36), 

77,  =  B  r1  log,  ^  =  R  r1  logj', 
and  for  the  heat  rejected  along  CD, 

H,  =  ErJ0gl±  =  RrJogj', 

and  for  that   absorbed   along  D  A  from   the  regenerator, 
equation  (37), 

S3  =  Cv  (r,  -  ra), 

and  rejected  along  B  C, 

H,  =  Ov  (r,  -  r2). 

The  heats  H3  and  ZT4  cancel  each  other.  In  practice  it  is 
found  that  a  certain  amount  of  energy  is  lost  in  the  regener- 
ator, as  stated  on  page  167,  which  we  represent  by 

n  Cv  (r,  -  r2). 


228  HEAT   ENGINES.  [120.] 

The  heat  transmuted  into  work  per  pound  of  working 
air  per  revolution  will  be 

U=Hl  +  Ht-  //,  -  //,  -  122.5  (r,  -  r,}  logiar,    (201) 
foot-pounds. 

If  the  regenerator  were  perfect,  the  efficiency  would  be 


which  is  the  same  as  that  of  the  perfect  elementary  engine, 
Allowing  for  the  imperfection  of  the  regenerator,  and  let- 
ting U'  be  the  work  actually  performed  by  the  motor  per 
pound  of  air,  we  have 

E=  _  v 

y/,  +  131»(r,-r,) 

The  dimensions  of  an  engine  must  be  found  in  design- 
ing it. 

Let,  in  Fig.  53, 
p,  with  the  subscript  of  the  letter  at  a  corner  of  the 

diagram,  represent  the  pressure  at  that  corner  ; 
v,  with  the  same  subscript,  represent  the  volume  ; 
r^KB-^RA  =  ratio  of  expansion  of  the  working 

air  ; 

q  =  R  E-*-  R  A  =  ratio  of  the  total  volumes  in  the 
cylinders,  passages,  and  clearances,  when  the  work- 
ing piston  is  on  its  lower  dead  centre  to  the  least 
volume  of  working  air. 

All  the  volumes  have  reference  to  one  pound  of  working 
air. 

We  have  K  B  =  r  .  R  A, 

or,  vb  =  rv&,  (203) 


E=(q-l}v>',  (204) 

=  pb  vb,     (2),  p.  11,  \ 


[120.] 


THEORY   OF   STIRLING'S   ENGINE. 


229 


If  *  be  'the  ratio  of  the  mass  of  working  air  to  that  of 
the  cushion  air,  and  r^  the  absolute  temperature  of  the  iso- 
thermal F  J,  we  have 

E  r%  =  pt .  s  vt  =  pK  .  s  VK  —  pi  .  s  Vi,  &c.  (206) 

From  these  we  find — 


Volumes  per  pound  of  the  working  air — 

v&  =  vi  =  53.21  -^-; 

^  (210) 

W  T 

vb  =  vc  =  rv,,=  53.21-^- 
P*    \ 

Volumes  of  cushion  air,  per  pound  of  working  air, 
vt  =  A  E  =  (<i  —  1)  wa  =  q~  1  vb,      (204),  (210),.   (211) 
•y    =  rv{    =  (q  —  l)v^  (212) 


=  -     vf  =     *-  v{  =  (q  - 

P\  PC 

v  <  —  1 


Total  volumes,  — 


=  ~ 


(206),  (207).     (213) 
(214) 


>       (215) 


230  HEAT   ENGINES.  [121.] 

Also 

Mass  of  cushion  air v{    _  r, 

Mass  of  working  air        -yc  rt 

Volume  swept   through,  by  the  piston  per  pound  of 
working  air  per  stroke — 

v,  -  ve  =  [l  +  (q  -  1)  -^  --  -£-]  vb-  (217) 

If  there  be  given,  instead  of  the  ratio  of  expansion,  the 
ratio  of  the  volume  swept  through  by  the  working  piston  to 

that  swept  through  by  the  plunger,  or  —3 2_,  then  r  may 

be  found  from  equation  (217),  giving 

r= L.— -.  (218) 


If  q  is  not  given,  find  an  approximate  value  of  r  by  mak- 
ing q  =  r,  giving 

r  =  !±_H^_  .  _!-_  +  i  .  (219) 


the  correct  value  of  which  will  be  somewhat  less  than  the 
value  thus  found,  and  q  will  be  somewhat  greater.  Assume 
q  about  2  to  5  per  cent  more  than  the  value  of  r  found 
from  (219),  and  find  the  correct  value  of  r  from  equation 
(218). 

If  the  piston  remained  on  its  dead  points  while  the 
plunger  was  moving,  and  the  plunger  on  its  dead  point 
while  the  piston  was  moving,  the  indicator  diagram,  and  the 
diagram  representing  the  changes  in  the  working  fluid, 
would  be  as  shown  in  Fig.  54.  The  approximate  analysis  of 
this  case  presents  no  serious  difficulty. 

121.  Ill  designing  an  engine  of  the  Stirling  type, 
the  horse-power  to  be  delivered  and  the  number  of  revolu- 


[121.]  IN   DESIGNING.  231 

tions  per  minute  must  be  known,  in  addition  to  the  data  al- 
ready assumed.  The  number  of  revolutions  will  be  limited 
by  the  piston  speed  and  the  length  of  stroke.  The  average 
piston  speed  may  be  between  100  and  200  feet  per  minuted 
One  of  Stirling's  engines,  having  a  four-foot  stroke,  was  run, 
in  actual  practice,  at  about  28  revolutions  per  minute,  giving 
an  average  piston  speed  of  about  221  feet  per  minute. 

An  air  engine,  reported  upon  by  M.  Tresca,  had  a  stroke 
of  0.4  m.  (1.3  ft.)  and  made  about  90  revolutions  per  min- 
ute, giving  a  piston  speed  of  about  120  feet  per  minute. 

The  large  air  engines  in  the  steamer  Ericsson  had  an 
average  piston  speed  of  108  feet  per  minute. 

Let  ]¥  =  the  number  of  revolutions  per  minute, 
S  =  the  average  piston  speed, 
I  =  the  length  of  stroke  of  the  piston, 
h  =  number  of  horse-power  required  of  the  engine, 
TF"  =.  the  work  required  of  the  engine  per  minute  ; 
then, 

S=2JVl  (220) 

IF  =33000  A.  (221) 

Let  w  =  the  number  of  pounds  of  working  air  required  ; 
then,  since  the  work  done  by  one  pound  per  revolution  will 
be  theoretically,  the  value  of  U  in  equation  (201),  we  have  : 

'     :    ' 


But  the  actual  work  U  will  be  less  than  the  theoretical, 
and  we  will  assume  it  to  be  0.7,  the  theoretical.  (In  de- 
signing it  is  better  to  assume  too  small  a  fraction  rather  than 
too  large.)  Then 

33000  h 
~-   0.7  X  122.5  (r.-.rOfag.rX^' 

If  r  be  assumed,  the  weight  of  air  in  one  cubic  foot  will 
be,  (205),  (210),, 


282  HEAT   ENGINES.  [121.] 

(223) 

The  initial  pressure,  pw  may  be  assumed,  since  it  can  be 
produced  and  maintained  by  the  air  pump  in  connection 
with  the  heat  derived  from  the  furnace. 

The  volume  of  the  lower  part  of  the  recewer  will  be, 
(223),  (222), 

— —  w  cu.  ft.  (223a) 

Assume  the  stroke  of  the  plunger  to  be  y  I,  in  which  y  is 
a  fraction,  say  £ ,  f ,  or  ^  ;  and  A  its  area  ;  then 

I    A  -   Rrr*   w 

.'.  A  =  53.21  !!JJJf.  (224) 

''/  '"  PA 

The  volume  swept  through  by  the  piston  per  pound  of 
working  air  per  stroke  being  given  by  equation  (217)  and 
the  stroke,  Z,  having  been  assumed,  we  have  for 

the  section,  B,  of  the  working  cylinder,  =    q     — -  w, 


(225) 
in  which  q  will  exceed  r,  and  will  be  assumed. 

If  -ll— i.  be  given,  instead  of  r,  first  find  r  approxi- 

^b 

mately  by  equation  (219) ;  then  assume  q  greater  than  r,  and 
find  r  by  (218),  after  which  proceed  as  before. 

The  mean  effective  pressure  for  the  single-acting  engine 
is  such  an  uniform  pressure  as  would,  if  acting  throughout 
the  upward  stroke,  do  the  same  work  as  is  done  by  the  fluid 
during  one  revolution  of  the  engine.  Since  w  pounds  of  air 


(121.  ]  IN   DESIGNING.  233 

does  the  work  w  U  during  this  time,  we  have 


EXERCISES. 

1.  Let  T,  =  600°  F. ;  T,  =  120°  F. ;  p&  =  120  Ibs.  per  sq. 
in.  ;  stroke  of  working  piston,  2£  feet ;  30  revolutions  per 

q\       qj 

minute  ;  — —    —  =  ^  =  the  ratio  of   piston   displacement 

vb 

to  plunger  displacement ;  and  5  horse-power  be  developed. 
Find 

rl  =  1060°  ;  r^  =  580°,  omitting  decimals. 

_!L  =  1.83;  -^-  =  0.55,  nearly. 

r  =  1.275,  approximately,  (219). 
Assume  q  =  1.30  ; 

then  r  =  1.25,  (218). 

U  =  5693  ft.-lbs.,  (201). 
E  =  0.453,  (202). 
E'  =  0.317,  if  0.7  E. 
2\  =  17280  Ibs.  per  sq.  ft.  (given). 
vb  =  4.08  cu.  ft,  (2 10)a. 
vq  —  ve  =  2.04  feet. 

Pounds  of  air,         w  =  1.0,  nearly,  (222),  theoretical, 
or,  w  —  1.43  Ibs.,  (222«),  practical. 

2  336 
Section  of  plunger,  A  ~  —     —  sq.  ft.,  (224)  ;  and  if  y  =  •£, 

then 

Diameter  of  plunger  =  2.46  feet. 

Section  of  piston  B  =,  1.17  sq.  ft.,  (225). 
Diameter  of  piston  B  =  1.23  feet. 
Mean  effective  pi^essure,  p^  =  2755  Ibs.  per  sq.  ft.,  (226). 


234  HEAT  ENGINES.  [122.] 

Find  also  the  numerical  values  of  vm  %,  vw  uft  vp  p^ 
and  pv 

2.  In   a   double-acting   engine   made    by    Stirling,  hav- 
ing a  piston    16   inches  diameter  and  a  stroke  of  4  feet, 
making  28  revolutions  per  minute,  it  was  found  by  calcula- 
tion, and  also  by  means  of  a  friction  brake,  that  the  work 
done  per  minute  on  the  piston  was  1670000  foot-pounds. 
There   were   passed    through  the  refrigerator    250  Ibs.  of 
water  per  minute,  and  its  temperature  was  increased  18°  F. 
while  passing.     Assuming  that  the  heat,  except  that  doing 
mechanical   work,    was    absorbed    by    the    water    passing 
through  the  refrigerator ;  find  the  foot-pounds  of  heat  ab- 
stracted by  the  water,  the  efficiency  of  the  fluid,  provided 
seven  tenths  of  the  heat  were  abstracted  by  the  refrigerator ; 
the  horse-power  of  the  engine  ;  and  the  foot-pounds  of  heat 
unaccounted  for  by  the  absorption  of  heat  by  the  water  in 
the  refrigerator. 

3.  In  the  engine  described  in  the  preceding  Exercise,  83 
pounds  of  coal  were  used  per  hour,  possessing  an  estimated 
thermal  capacity  of  11580  B.  T.  U.  ;  find  the  efficiency  of 
the  plant  ;  also  of  the  furnace,  if  that  of  the  engine  be  0.3. 

Ans.  Efficiency  of  plant,  0.133. 
Efficiency  of  furnace,  0.44. 

From  this  it  will  be  seen  that  the  efficiency  of  the  furnace 
is  considerably  less  than  that  of  the  steam  boiler,  which,  in 
good  condition,  may  be  assumed  to  be  between  0.60  and 
0.75.  The  efficiency  of  the  plant  is  nearly  equal  to  that  of 
the  most  efficient  steam  plants  of  the  present  day.  See 
page  201.  But  exact  comparisons  cannot  be  made,  for  the 
thermal  capacity  of  the  coal  is  not  known  with  sufficient 
exactness,  nor  the  comparative  physical  properties  of  air  and 
steam  in  regard  to  conductivity. 

122.  Ericsson's  Engine,  in  which  the  changes  of 
temperature  are  made  at  constant  pressure.  About  the 
year  1833  John  Ericsson  constructed  in  London  a  so-called 


[123.  J  DESCRIPTION.  235 

"  caloric  engine,"  whicli  attracted  much  attention,  especially 
from  scientific  men ;  but  it  was  not  a  commercial  success. 
His  efforts  at  producing  large  engines  of  this  class  cul- 
minated in  making  in  New  York,  in  1853,  a  vessel  of  2200 
tons,  called  the  Ericsson,  in  which  the  motors  consisted  of 
four  immense  caloric  engines.*  (For  dimensions,  see  Exer- 
cise 1,  following.)  After  experimenting  with  these  weak 
giants — giants  in  size,  but  weak  in  power — they  were  aban- 
doned ;  but  he  produced  another  hot-air  engine,  f  which  was 
extensively  introduced  in  various  parts  of  the  world  ;  still, 
after  a  few  years,  many  of  them  were  removed  and  replaced 
by  steam  engines.  Their  great  bulk,  the  noise  attendant 
upon  their  working,  and  the  rapid  destruction  of  their  ,  fur- 
naces, were  prejudicial  to  their  general  use.  More  recently 
Captain  Ericsson  has  designed  a  small  hot- air  pumping 
engine,  which  is  being  extensively  used,  the  principles  of 
which  we  will  consider. 

123.  Description.  Fig.  55  is  an  external  view  of  a 
small  hot-air,  Ericsson  pumping  engine,  and  Fig.  56  is 
a  sectional  view  of  the  same.  Within  a  cylinder  of  uni- 
form bore  are  two  pistons,  A  and  B,  of  which  B  is  the 
driving  piston  and  operates  the  mechanism  in  a  manner  so 
clearly  shown  as  not  to  need  explanation.  The  lower  piston, 
A,  which  we  will  generally  call  the  plunger,  is  made  of 
some  substance  which  is  practically  a  non-conductor  of 
heat.  Its  office  is  to  transfer  a  body  of  air  from  the  space 
below  it  to  the  space  above,  and  back  again,  and  so  on  alter- 
nately, and  for  this  reason  is  known  as  the  displacing,  or 
transferring,  piston.  In  the  position  shown,  the  plunger  A 
is  at  the  upper  end  of  its  stroke,  and  the  piston- 7?,  being 
governed  in  its  speed  by  the  crank  I,  is  moving  most  rapidly, 
and  is  driven  by  the  expansion  of  the  air  in  the  lower  part  of 

*  Journal  of  Arts  and  Science,  Sept.,  1883. 

f  Contribution  to  the  Centennial  Exhibition,  1875,  by  John  Ericsson, 
pp.  425-38. 


236 


HEAT   ENGINES. 


[123.] 


the  receiver  fl.  The  plunger  remains  nearly  stationary,  de- 
scending but  little  while  the  piston  completes  its  upward 
stroke.  The  air  in  the  upper  part  of  the  receiver  is  cooled 
by  the  water  which  has  been  raised  by  the  pump  r  circulat- 
ing in  the  annular  space  xx,  so  that  the  water  raised  for 
other  useful  purposes  acts  as  a  refrigerator  of  the  engine. 
During  the  earlier  part  of  the  return  stroke  of  the  piston, 
the  plunger  descends  a  little  faster  than  the  piston,  main- 


FIG.  55. 


FIG.   56. 


taining  a  nearly  uniform  pressure  upon  the  piston  while  air 
is  transferred  from  below  the  plunger  to  the  space  above, 
the  volume  and  temperature  both  decreasing.  When  the 
piston  has  reached  about  the  position  shown  in  Fig.  56  on 
its  downward  stroke,  the  plunger  will  have  reached  the 
lower  end  of  its  stroke  and  all  the  working  air  will  have 
been  transferred  above  and  its  temperature  maintained  at 
its  inferior  limit  while  it  is  compressed  by  the  completion 


[124.]  ANALYSIS.  237 

of  the  downward  stroke  of  the  piston  JB ;  after  which  the 
plunger  will  rise  to  the  position  assumed  at  the  beginning  of 
this  description,  during  which  the  working  air  will  be  trans- 
ferred to  the  lower  part  of  the  receiver,  and  its  temperature 
and  volume  both  increased  at  nearly  constant  pressure. 
The  mass  of  air  in  the  engine  is  constant. 

1134.  Analysis.  Fig.  57  is  a  copy  of  an  indicator 
diagram  taken  from  a  small  engine  of  this  class  in  Stevens 
Institute  of  Technology.  It  will  be  seen  that  the  changes 
of  temperature  at  constant  pressure  are  clearly  indicated, 


FIG.    57. 

and  the  isothermals,  being  nearly  straight  lines,  show  that 
the  variation  of  pressure  is  small  compared  with  the  change 
of  volume. 

Assuming  that  the  change  of  state  from  that  of  constant 
pressure  to  that  of  constant  temperature  is  instantaneous, 
the  diagram  of  one  pound  of  the  working  fluid  may  be  rep- 
resented by  D  E  F '  G,  Fig.  58.  F  will  represent  the 
state  of  the  working  fluid  at  its  highest  temperature,  r]? 
greatest  volume  and  least  pressure ;  hence  the  plunger  and 
piston  will  both  be  at  the  upper  ends  of  their  strokes  in  the 
ideal  case ;  and  the  mass  of  air  below  the  plunger  will  be 
considered  as  working  air,  and  all  the  other  air  cushion  air. 
I  -Z^will  represent  the  volume  of  one  pound  of  working  air 
at  its  highest  temperature,  and  corresponds  to  the  space 
below  the  plunger.  Let  F  Fi  to  the  same  scale  corre- 
spond to  all  the  space  above  the  working  air ;  then  will 


238 


HEAT  ENGINES. 


1124.] 


correspond  to  the  entire  volume  of  the  cylinder  per 
pound  of  working  air.  The  cushion  air  being  supposed 
to  remain  at  the  inferior  limit  of  temperature,  take  /  L  = 
F  FI  and  construct  the  isothermal  L  K  for  the  temperature 
T2,  to  represent  the  path  of  the  cushion  air.  Make  D  D,  = 


A         B 


T ._—•• 


H  K  =  E  Et,  G  G,  =  I L  ;  then  will  D,  E,  F,  G,  be  the 
real  indicator  diagram  of  the  engine.  If  Dl  falls  to  the 
left  of  F,  the  piston  at  the  lower  end  of  its  stroke  will  pass 
into  the  space  occupied  by  the  working  air  at  its  greatest 
volume. 

Let  rt  be  the  absolute  temperature  of  the  isothermal  E  F, 
and  TS  that  of  D  G,  and  Cp  the  dynamic  specific  heat  of  air 
at  constant  pressure;  then  will  the  heat  absorbed  per 
pound  of  air  be,  from  state  D  to  state  E, 

along  E  F, 

along  F  G, 

along  GD, 

—Ht  =  —  R  T,  log,,  r ; 

hence,  the  work  done  per  pound  of  air  per  revolution,  if 


[124.  J  ANALYSIS.  239 

the  conditions  were  perfect,  would  be  the  sum  of  these,  or 
U  =  122.5  (r,  -  T.)  %10r.  (227) 

Efficiency  of  fluid  with  perfect  regenerator  y 

E=^<  (228) 


or  the  efficiency  would  be  the  same  as  that  of  the  perfect 
elementary  engine.  There  being  no  regenerator,  the  effi- 
ciency of  fluid,  if  working  perfectly  without  radiation,  will  be 

(228«) 

If  all  the  losses  due  to  radiation  and  the  refrigerator  be 
represented  by  n  Cv  (^  —  rj,  then  the  efficiency  would  be 


TJ    .    ,Q.      . 
Hl  -f-  184r7i(r,  —  r2) 

The  value  of  n  is  not  known,  but  will  exceed  unity  in  this 
class  of  engines,  especially  with  very  slow  speed. 

In  this  analysis,  the  pressure  at  state  G  will  be  assumed 
to  equal  that  of  the  atmosphere,  although  it  may  be  some- 
what less,  as  shown  in  Fig.  57  ;  then  if 

Pi  be  the  pressure  per  square  foot  of  the  atmosphere,  ra 
its  absolute  temperature,  and  v&  the  volume  of  a  pound  ; 
then 

p*  v*  =  R  ra,  (2)  ;  (229) 

.•.-»=  53.21  -^~ 


and  -w,  with  subscripts,  as  in  Article  120,  represent 
respectively  the  pressures  and  volumes  at  the  corresponding 
states  ;  then 

A  =P^    P**t  =  XT,;  (230) 


240  1IKAT    FNUIXKS.  [134.] 

from  which  it  appears  that  if  r^  exceeds  ra,  VK  will  exceed 
•ya.  v&  may  be  taken,  roughly,  at  12£  cubic  feet.  If  r  be  the 
ratio  of  expansion,  we  have,  making  r  —  r2  in  Eq.  (2), 

^_=J^_  =  r=  *L=:J!l=_rL.  (231) 

vd         j}K  pt         ve         vk 

From  the  figure  and  equation  (231),  we  have 
Pressures  — 

Pt=l>g=  Pt,  =  pgl  ;    pA=  p*  =  PK  =  pn  =  rpg.   (232) 
Volumes  —  working  air, 


Cushion  air  ;  let  q  be  the  ratio  of  the  entire  volume  of  air 
in  the  cylinder  to  the  greatest  volume  of  working  air,  then 

9l  =  vn  —  v,  =  (q  —  1)  vr  ;        t-k  =  A  =  -?-=-  ~  r.  ;  (234) 

Total  volumes— 


(235) 


r          r 

Volume  swept  through  by  the  piston  per  pound  of  work- 
ing air  per  revolution — 

2  K  -  vdl)  =  2  ( q  (r  -  1)  +  1  -  -I«-\  ^L.        (236) 


Mean  effective  pressure  per  unit  area  of  the  piston,  or 
energy  expended  per  foot  of  volume  swept  through  by  the 
piston,  distributed  over  two  strokes — 

r.   = 5L  (237) 

-^       2  K  -  «dl) 


[124.]  ANALYSIS.  241 

Mean  total  forward  pressure — 


Mean  back  pressure — 

PB  =  P*  -  pv  (239) 

Greatest  vol.  working  air  vf 

Piston  displacement  vfl  —  vA' 

Let 

.ZT  be  the  number  of  revolutions  per  minute, 
/$,  the  average  piston  speed, 
Z,  the  length  of  stroke  of  the  working  piston, 
TF,   the  work  in  foot-pounds   developed  by  the  piston 

per  minute, 

HP,  the  horse-power  developed  per  minute, 
w,  the  pounds  of  working  air  per  revolution, 
A,  the  area  of  the  working  piston  ; 
then 

S=  2  Nl; 

W  =  33000  HP ;  (241) 

TF  =  w  U  N  =2pflA  N.  (242) 

If  the  isothermals  are  so  nearly  right  lines  that  they  may 
be  considered  as  straight,  the  indicator  diagram  may  be 
treated  as  a  trapezoid ;  hence,  its  area,  referring  to  Fig.  58, 
will  be 

GFx  H 1=  G,F,  x  ///; 

or,     (vt  -  O  (pA  -  p,}  =  53.21  (T,  -  r,)  (r  -  1),      (243) 
for  the  work  done  per  pound  of  air  per  revolution. 

EXERCISES. 

1.  In  the  steamer  Ericsson,  there  were  four  single-acting 
working  cylinders,  producing  an  aggregate  of  300  horse- 
power, as  determined  by  an  indicator.  The  pistons  were 
14  feet  in  diameter ;  length  of  stroke,  6  feet ;  revolu- 


242  HEAT   ENGINES.  [124.J 

tions,  9  per  minute ;  fuel,  1.87  pounds  of  coal  per  horse- 
power per  hour,  the  total  heat  of  combustion  of  each  pound 
being  estimated  at  MOOO  thermal  units. 

Let 

T,  =  420°  F.,  Ts  =  120°  F. 

From  this  data  find— 
Mean  eff.  pressure,  Ibs.  per  sq.  in.  per  double  stroke. .     1.1. 

Average  piston  speed,  feet  per  minute 108. 

Vol.  swept  through  by  piston  per  I.  H.  P.,  ft.  per  min.  222. 

Heat  of  combustion  of  one  Ib.  coal 

14000  X  778  ft.-lbs.  =  10892000. 
Duty,  1  Ib.  coal.. 33000  X  60  -j-  1.87  ft.-lbs.  =    1059000. 

Efficiency  of  plant..   .^™  *J^  ,        0.097!. 

Theoretical  efficiency  of  fluid,  (228) E  =         0.340. 

Actual  efficiency  if  0.8  of  theoretical 0.272. 

Probable  efficiency  of  furnace.0.0971  -=-  0.272  =         0.357. 

Notwithstanding  the  good  efficiency  of  the  plant,  the  ex- 
cessive size  of  the  cylinders  and  other  practical  considera- 
tions prevented  the  general  introduction  of  this  class  of 
engines  for  large  powers. 

2.  Let  the  bore  of  the  air  cylinder  of  the  pumping  engine 
shown  in  Figs.  55  and  56  be  6  inches,  stroke  of  the  working 
piston,  B,  2£  inches,  stroke  of  the  plunger,  0,  5|  inches, 
diameter  of  the  plunger,  5f  inches  (length  of  the  plunger 
about  20  inches),  being  the  dimensions  of  this  engine  in 
the  Institute.  The  piston  at  the  lower  end  of  its  stroke 
passes  into  the  plunger  space  about  one  inch,  and  near  the 
middle  of  the  stroke,  as  shown  in  Fig.  56,  there  is  about  TV 
of  an  inch  between  the  piston  and  plunger,  thus  re- 
ducing the  cushion  air  to  a  minimum.  The  furnace  extends 
upward  about  one  half  the  length  of  the  plunger,  above 
which  the  cushion  air  surrounds  the  plunger  when  at  the 
upper  end  of  its  stroke ;  and  as  only  that  is  effective  work- 


[124.] 


ANALYSIS. 


243 


ing  air  which  is  subjected  l>oth  to  the  refrigerator  and  fur- 
nace, the  working  air  will  be  less  in  volume  than  that  of  the 
plunger-  displacement  ;  but  the  relation  cannot  be  deter- 
mined with  accuracy.  As  nearly  as  we  can  determine  in 
this  engine,  we  have. 


=  0.75,  Eq.  (240). 


vf 


Assume  ra  =  520  ;  T,  =  130°  F.  ;  T,  =  720°  F.  ;  total 
air  volume  per  pound  of  working  air  at  its  greatest  vol- 
ume, q  =  1.2;  and  50  revolutions  per  minute. 

Find  :— 


r,  =  590  ;  rt  =  1180  ;  rl  —  r^  =  590 


-  =  0.5. 


Greatest  vol.  of  a  pound  of  working  air, 

(233),  cn.it  ........................  vf  =  29.67. 

Greatest  total  volume,  (235),  ..........  '  vtl  =  35.60. 

Least  total  volume,'  (235),,  or  (243«).  .  .  .  vdl  =  11.35. 

Volume  swept  through  by  the  piston  per 

pound  of  working  air  per  stroke,  (243#), 

cu.  ft  ............................  0fl  —  vd,  =  24.25. 

Katio  of  expansion,  (235),  or  (236)  ......  r    =       l.S. 

Work  per  Ib.  of  air  per  revolution, 

(227),  ft.-lbs  ......................  .  U  =  18450. 

M.E.  P.,  (237),  (2430),  (233),,  Ibs.  per 

sq.  ft.  double  stroke  ................  j?>«  =  414.6. 

M.  E.  P.  for  the  single  working  stroke 

of  air,  Ibs.  per  sq.  ft  ................  p^  —  829.2. 

M.  E.  P.,  for  the  single  working  stroke 

of  air,  per  sq.  in  ....................  P*  =    5.76. 

Area  of  working  piston,  sq.  in  ..........  28.2744. 

Work  per  revolution,  ft.-lbs.  .  28-27^  x  2j   829.7  =  33.92. 

Work  per  minute,  ft.-lbs  .............  50  X  33.92  =  1696. 

Horse-power  ...........  .........  1696  -r-  33000  =  0.051. 


244  HEAT    ENGINES.  [124.] 

Efficiency,  if  n  =  1  in  Eq.  (228a)  ......  E  =  0.128. 

Efficiency  of  plant  if  eff.  of  furnace  be  0.3  0.037. 

Pounds  of  working  air,  (241)  ..........  w  =  0.0018. 

The  pump  described  in  this  Exercise  is  used  by  the 
students  in  their  experimental  course,  and  from  one  of 
those  I  make  the  following  abstract  ; 

At  50  rev.  per  m.  indicated  M.  E.  P.  was,  Ibs.  per  sq.  in  .  .  5.42. 

^  .        5.42  X  28.2744  X  2|  X  50 
1.  work  perm.,  it.-lbs.  .—  —  ^  —  -  =  1597. 

Indicated  horse-power  ..........  1597  -f-  33000  =  0.0484. 

Weight  of  gas,  cu.  ft.  per  hour  .....................  15.7. 

"Weight  of  one  cubic  foot  of  the  gas  .............  0.04584. 

Calorific  power,  E.  T.  U.  per  Ib  ..................  13650. 

"  "        "    "    "     "    cu.  ft  ................  625.7. 


Ind.ef.  of  ft—  ,d  fluid,  157       627x  778  =  0.0125, 

or  about  1^  per  cent. 

This  result  shows  the  great  loss  of  heat  in  this  engine 
without  a  regenerator.  Either  the  furnace  has  an  efficiency 
of  only  about  0.1,  or  if  its  efficiency  be  0.3,  then  n  in  equa- 
tion (228«)  should  be  between  3  and  4.  If  the  efficiency 
of  the  furnace  be  about  0.2,  then  n  must  be  2  or  more. 

The  effective  power  as  determined  by  the  water  pumped 
was  640  foot-pounds  per  minute  ;  hence  the  loss  by  leakage 
and  friction  was  estimated  to  be 
1597  -  640  _ 

1597 
or  nearly  60  per  cent. 

Efficiency  of  plant,  including  fuel,  engine  and  pump 

' 


or  only  one  half  of  one  per  cent  of  the  theoretical  heat  of 
combustion  of  the  fuel  was  utilized  by  the  plant.  "  This  was 
one  of  the  best  of  fifteen  experiments. 


[125.]  RATIO   OF    EXPANSION.  245 

The  manufacturers  guarantee  that  this  size  of  pump  will 
raise  200  gallons  of  water  per  hour  50  feet  high  with  18 
cubic  feet  of  gas.  This  would  give  an  effectual  work  of 

200  X  5^-  X  —  X  62  £  =  1383  foot-pounds.    This  is  more 

60 


than  twice  the  amount  found  by  the  experiment  above 
cited  ;  but  a  part  of  the  difference  may  be  due  to  the  fact 
that  more  gas  is  required  than  was  consumed  in  the  experi- 
ment, the  quality  of  the  gas,  the  condition  of  the  engine, 
etc.  ;  and  the  remainder  —  if  any  —  to  the  art  of  advertising. 

A  very  small  power  steam-engine  with  furnace  and  boiler 
may  require  from  8  to  12  pounds  of  coal  per  horse- 
power per  hour,  giving  an  indicated  efficiency  of  some  2 
per  cent,  more  or  less.  The  hot-air  pumping  engine  is  used 
not  on  account  of  its  superior  efficiency,  but  on  account  of  its 
greater  economy  and  safety—  there  being  no  danger  of  ex- 
plosion, and  requiring  but  little  expense  for  attendance. 

125.  Ratio  of  expansion  to  give  a  maximum 
mean  effective  pressure.  A  general  solution  cannot  be 
made.  We  will  assume  some  elements,  and  thus  illustrate 
the  process  for  a  particular  case. 

Let  q  =  1.3  ;  Vfl~  ^dl  =  0.8  ;  m  =  -'  ;  and  let  12£  cubic 

V(  T1 

feet  of  air  weigh  a  pound. 
Then,  (236) 

m  =  0.5  r  —  0.3. 

With  these  conditions,  and  equations  (227)  and  (237), 
we  have 

122  5 
P*  =  -^-  ^  (1.3  -  0.5  r)  logj,  (244) 

which  is  a  maximum  for  r  =  1.69,  as  may  be  found  by 
trial.  The  corresponding  value  of  m  will  be 

m  =  0.545. 


246  HEAT   ENGINES.  [126.] 

The  value  of  p,  in  (244)  will  be  zero  for 

'  -  =  £§  =  8.6,  .  . 

under  which  condition,  the  engine,  if  f  rictionless,  would  run 
without  doing  work,  and  would  simply  change  the  states  of 
the  working  fluid.  The  mean  effective  pressure  is  unaffected 
by  the  initial  temperature ;  simply  the  range  of  tempera- 
tures being  involved  in  the  value  of  m,  which  may  finally 
be  expressed  as  a  function  of  r,  as  above.  The  work  per 
minute  will  depend  upon  the  number  of  revolutions,  and 
heat  must  be  supplied  in  sufficient  quantity  and  with  suffi- 
cient rapidity  to  maintain  the  assumed  temperatures. 

REMARK. — The  application  of  the  analysis  in  the  two  preceding  articles 
is  very  delicate,  for  there  are  so  many  physical  conditions  that  cannot 
be  definitely  determined,  and  a  small  change  in  any  one  of  them  may 
produce  a  large  change  in  the  results.  Thus,  it  will  be  seen  that  changing 
the  value  of  q  from  1.2,  as  in  the  preceding  Exercise,  to  1.8  in  this 
Article,  changes  r  from  1.75  to  1.69.  (The  value  used  in  the  Ex 
ercise  is  1.8,  being  the  nearest  entire  tenth,  but  its  value  is  exactly  1.75.) 
The  "log  r"  will  be  changed  in  a  greater  ratio,  thus  affecting  the  final 
result  in  a  corresponding  manner.  The  solution  given  shows  that  the 

fraction  —     —  is  large  when  the  engine  is  run  at  its  best  effect ;  and  if 

it  be  assumed  as  0.5  it  would  produce  much  less  work  per  minute.  The 
greater  the  difference  of  the  temperatures  the  greater  the  work  done  per 
pound  of  working  air,  and  the  greater  the  ratio  of  the  absolute  tempera- 
tures the  greater  the  efficiency,  other  things  being  equal. 

The  working  of  the  engine  is  quite  as  delicate  as  the  analysis,  it  being 
much  affected  by  friction  in  the  cylinder  and  stuffing  boxes,  and  the  con- 
dition of  the  furnace. 

126.  Heat  received  and  rejected  only  at  con- 
stant pressure.  An  engine  involving  this  principle 
was  proposed  by  Joule  and  Thomson  (Phil.  Trans.,  1885), 
but,  so  far  as  known,  has  not  been  constructed.  In  this 
engine  the  expansion  and  compression  of  the  fluid  would 
be  adiabatic.  A  B  C  D,  Fig.  59,  page  238,  would  be  the 
indicator  diagram  of  such  an  engine,  in  which  A  B  and  D  C 
are  parallel  to  the  axis  O  v,  and  B  C  and  A  D  are  adiabatics. 


[127.]  A   GAS   ENGINE.  247 


GAS  ENGINES. 

.  A  gas  engine  is  a  hot-air  engine  in  which  the 
cylinder  containing  the  working  air  is  also  the  furnace,  heat 
being  produced  by  the  rapid  combustion  of  the  fuel  in  the 
cylinder— so  rapid  as  to  be  called  an  explosion.  The  fuel 
is  an  inflammable  gas.  When  the  piston  is  moving  for- 
ward in  its  stroke,  air  and  gas  are  drawn  into  the  cylinder, 
and,  at  the  proper  time,  the  gas  is  ignited,  an  explosion 
takes  place,  the  air  is  suddenly  heated  and  a  high  pressure 
produced  ;  after  which  a  part  of  the  energy  thus  developed 
is  imparted  to  the  piston  during  the  remainder  of  the  stroke, 
and  the  other  part  is  forced  out  of  the  cylinder  at  the  ex- 
haust. 

The  two  most  prominent  systems  which  have  been  de- 
veloped are  :  one  in  which  the  charge  is  fired  with  every 
revolution,  when  the  cylinder  is  about  half  full  of  air  and 
gas  ;  the  other  at  each  alternate  revolution,  when  the  piston 
is  near  its  remote  dead  point.  In  the  former,  the  energy 
developed  can  act  on  the  piston  during  only  about  one  half 
of  a  single  stroke  ;  while  in  the  latter  it  will  act  during 
nearly  the  whole  stroke ;  so  that  the  latter  ought  to  be,  as  it 
is  found  to  be  in  practice,  much  more  efficient  than  the 
former.  Fig.  60,  page  238,  illustrates  an  ideal  diagram  of 
the  former  engine. 

In  nearly  all  the  more  recent  gas  engines  the  piston 
draws  in  the  charge  of  gas  and  air  during  a  full  forward 
stroke,  then  compresses  it  during  the  next  backward  stroke ; 
and  when  just  past  the  next  dead  point  the  gas  is  ignited 
and  the  piston  is  driven  by  the  energy  thus  developed  dur- 
ing the  next  forward  stroke,  and  during  the  next  backward 
stroke  the  products  of  combustion  are  forced  out ;  thus  requir- 
ing two  revolutions  to  complete  a  cycle.  These  are  trunk 
engines.  Gas  engines  are  made  which  take  a  charge  at 


248  HEAT    ENGINES.  [128.] 

both  ends  of  the  cylinder  and  thus  resemble  double-acting 
engines,  although,  in  reality,  there  is  only  one  explosion 
during  each  revolution.  Others,  like  the  Clerk  engine, 
compress  the  charge  in  an  auxiliary  cylinder  which  is  fired 
in  one  end  of  the  working  cylinder  with  every  revolu- 
tion. Thus,  while  the  steam-engine  has  been  improved 
by  passing  from  single  acting  to  double,  quadruple,  &c., 
acting  during  each  revolution,  the  gas  engine  has  been  im- 
proved by  passing  from  double  to  single  acting  during  each 
revolution,  and,  finally,  to  one  action  during  a  bi-revolution. 
128.  History.  The  origin  of  the  gas  engine  is  not 
definitely  known.  It  appears  to  be  an  outgrowth  of  an  effort 
to  use  gunpowder  as  the  fuel,  which  substance  was  sug- 
gested for  this  purpose  as  early  as  1680  by  the  celebrated 
Huyghens.  The  gas  engine  proper  was  first  patented  in 
England  more  than  a  century  later,  1794,  and,  although  in 
the  years  following  there  were  many  improvements  and 
many  patents,  yet  it  became  of  no  practical  value  until  about 
I860,  during  which  year  M.  Lenoir  constructed  in  Paris  the 
first  gas  engine  that  was  actually  introduced  into  public  use  ; 
and  during  the  five  years 
immediately  following  several 
hundred  were  used  in  France. 
It  was  patented  in  England 
by  J.  II.  Johnson.  It  was  of 
the  non-compression  type,  and 
in  its  external  appearance  re- 
sembled the  ordinary  double- 
acting  steam-engine.  The  charge 
was  fired  at  each  end  during 
each  revolution.  It  contained 

no  new  principle,  and  its  success  was  the  result  of  the  care 
and  thoroughness  with  which  the  details  were  worked  up. 
A.  section  is  shown  in  Fig.  61.  Fig.  62  is  an  indicator 
diagram  taken  from  a  two  horse-power  engine  of  this  class, 


(128.J 


HISTORY. 


249 


FIG.  62. 


as  shown  in  the  Journal  of  the  Franklin  Institute^  Vol. 

LI.,  1866,  Feb.,  p.  176. 

The  length  of  the  line  A  £,  Fig.  62,  represents  the  length 

of  stroke  of  the  engine,  while  the  line  itself  is  the  atmos- 

pheric  line.      Three   lines    are   traced 

representing  the  action  at  one  end  of 

the    cylinder    during   six   revolutions. 

From  A  to  b  the  charge  was  taken  in  at 

atmospheric  pressure,  but  from  b  to  <?, 

the  inlet  valve  being  partly  closed,  the 

pressure  fell,  and  at  c  the  valve  closed,  the  charge  was  fired 

and  the  pressure  suddenly  raised  ;   and  the  energy  thus  de- 

veloped drove  the  piston  to  the  end  of  its  stroke.     During 

the  return  stroke  the  products   of  combustion  are  driven 

out  at  atmospheric  pressure. 

In   1867    Otto   and  Langen   exhibited   their  free  piston 

engine,  of  which  Fig.  63  is  an  external  view  of  one  of  this 
class  in  Stevens  Institute.  The  princi- 
ple was  not  new,  but  its  details  were  so 
well  worked  up  that  it  became  a  com- 
mercial success.  It  acts  by  drawing  in  a 
charge  of  air  and  gas  during  the  first 
few  inches  of  its  stroke,  then  the  valve 
is  closed,  the  charge  fired,  and  the  pis- 

Iton,  which  is  free,  is  shot  upward,  and  a 
jai  ^  partial  vacuum  formed  within  the  cylin- 

der, while  the  pressure  of  the  atmos- 
phere  on  the  piston  gradually  brings  it  to 
rest  and  then  forces  it  downward.  Dur- 

<  1  <  >  W  1  1  Wll  l'(  1        lllOt5(>M      R      pU\\'l     Oil      tllC 

piston  rod  engages  a  ratchet  on  the  main 
shaft,  thus  imparting  to  the  latter  a 

rotary  motion,  which  is  rendered  nearly  uniform  by  the  fly- 

wheel. 

The  idea  of  compressing  the  charge  before  explosion  was 


FIG.  63. 


250  HEAT   ENGINES.  [128.] 

mentioned  as  early  as  1801,  but  the  system  now  generally 
used  was  patented  by  Barnett,  an  Englishman,  in  1838,  and 
by  Million,  a  Frenchman,  in  1861,  and  further  developed 
by  M.  Beau  de  Rochas  in  France  and  Sir  C.  W.  Siemens 
in  England,  both  in  1862.  The  advantages  of  compression 
became  fully  recognized  by  this  time,  and  the  principle  has 
been  incorporated  into  nearly  all  gas  engines  constructed 
since  that  date. 

In  187G  M.  Otto  produced  his  "  Otto  Silent"  engine, 
which,  for  smoothness  and  quietness  of  running,  and  the 
economy  in  the  use  of  the  gas  fuel,  far  exceeded  all  pre- 


vious inventions  of  this  class  of  engines,  and  in  less  than  ten 
years  after  its  invention  it  is  claimed  that  15,000  were  sold. 
No  new  principle  was  incorporated,  the  success  being  en- 
tirely dependent  upon  the  skilful  use  of  the  principles  de- 
veloped by  others.  Fig.  6i  is  an  external  view  of  an  "  Otto,'' 
used  in  making  experiments  in  Stevens  Institute. 

Successful  gas  engines  of  many  varieties  are  now  used. 
At  the  American  Institute  Fair,  in  the  fall  of  1887,  six 
different  types  were  exhibited  by  as  many  different  invent- 
ors ;  among  which  was  an  "  Otto"  containing  the  most 
recent  improvements,  some  of  which  were  exceedingly 


[129.  J  SOME  DETAILS.  251 

ingenious,  and  a  "  Baldwin"  of  recent  invention,  which  for 
silent  running  and  uniformity  of  motion  seemed  to  be  all 
that  could  be  desired.  All  these  engines  outwardly  re- 
sembled the  modern  horizontal  steam-engine.  Some  en- 
gines of  this  class  are  duplex,  some  vertical,  but  mostly 
horizontal. 

The  great  improvement  made  in  the  gas  engine  is  strik- 
ingly illustrated  by  the  fact  that  the  first  successful  ones, 
Lenoir's,  consumed  about  100  cubic  feet  of  gas  per  indicated 
horse-power  per  hour,  while  an  Otto  has  consumed  less  than 
20  cubic  feet  for  the  same  power.  Some  of  the  earlier 
engines  consumed  more  than  100  cubic  feet,  and  the  latter 
ones  more  generally  require  about  24  cubic  feet.  The  low- 
est figure  given  above  was  for  a  rich  gas  and  an  8  H.  P. 
engine. 

129.  Some  details.  Between  the  piston  at  its  dead 
point  and  the  end  of  the  cylinder  is  a  space  not  swept  over 
by  the  piston,  called  the  combustion  chamber  ;  the  volume 
of  which  is  0.4,  more  or  less,  of  the  entire  volume  of  the 
cylinder. 

The  fly-wheel  is  large  compared  with  the  power  developed 
to  insure  more  uniform  running.  The  speed  is  also  regu- 
lated in  part  by  a  governor,  which  operates  differently  in 
different  engines.  In  some  it  cuts  off  a  part  of  the  supply 
of  gas  with  each  charge;  in  others  it  cuts  off  an  entire 
charge  until  the  speed  is  properly  reduced ;  and  in  still 
others  it  closes 'the  exhaust  so  as  to  retain  a  part,  or  all,  of 
the  products  of  combustion  of  the  previous  explosion,  thus 
preventing  a  full  charge  of  both  air  and  gas  being  taken  in. 

The  gas  is  ignited  in  various  wrays.  A  flame  of  gas,  ex- 
ternal to  the  cylinder,  communicating  with  the  interior  by  a 
small  orifice  covered  by  the  piston  until  the  charge  is  taken 
in  and  then  uncovered  during  its  regular  stroke,  has  proved 
to  be  efficient.  The  orifice  may  be  so  small  as  to  remain 
open  during  the  explosion,  but  in  the  more  recent  engines 


252 


HEAT    ENGINES. 


[129.] 


the  orifice  is  covered  by  a  valve.  The  flame  may  be  carried 
a  very  short  distance  in  a  cavity  of  a  valve.  Incandescent 
metal,  so  rendered  by  a  flame  or  by  an  electric  current,  has 
been  successfully  used.  In  some  cases  ignition  has  been  pro- 
duced by  an  electric  spark ;  and  in  still  others  by  chemical 
action.  This  is  an  exceedingly  important  detail,  and  has 
given  inventors  much  trouble,  as  its  action  must  not  only  be 
certain,  but  must  act  promptly  at  a  detinital  part  of  the 
stroke,  and,  sometimes,  more  frequently  than  90  times  per 
minute. 

A  space  is  provided  for  the  circulation  of  water  about  the 
cylinder,  through  the  piston,  and  also  through  the  cylinder 
heads.  This  is  rendered  necessary  to  prevent  injury  to  the 
metal  from  the  high  temperatures  due  to  the  explosion. 
Mr.  Dugald  Clerk  made  experiments  upon  gas  exploded  in 
a  closed  vessel,  determining  the  time  of  explosion  and  the 
pressures  resulting,  from  which  the  temperature  was  com- 
puted by  means  of  equation  (1),  page  11.  He  gives  the 
following  results  : 

MIXTURES  OF  AIR  AND  OLDHAM  COAL  GAS. 

TEMPERATURE   BEFORE   EXPLOSION,    17"  C. 


Mixture. 

Max.  press,  above 
atmos.  Inponnd* 
per  gq.  in. 

Temp,  of  explosion 
calculated  from 
observed  pressure. 

Theoretical  temp, 
of  explosion  if  all 
heat  were  evolved. 

Gas. 

Air. 

1  vol. 

14  vols. 

40 

806°  C. 

1786°  C. 

1vol. 

13  vols. 

51.5 

1033°  C. 

1912°  C. 

1  vol. 

12  vols. 

60 

1202°  C. 

2058°  C. 

1  vol. 

11  vols. 

61 

1220°  C. 

2228°  C, 

1  vol. 

9  vols. 

78 

1557°  C. 

2670°  C. 

1  vol. 

7  vols. 

87 

1733°  C. 

3334°  C. 

1  vol. 

6  vols. 

90 

1792°  C. 

3808°  C. 

1vol. 

5  vols. 

91 

1812°  C. 

1vol. 

4  vols. 

80 

1595°  C. 

I 

(The  Gas  Engine,  by  D.  Clerk,  p.  111.) 
The  computed  temperatures  may  not  have  been  the  highest 


[130. 


THEORY. 


253 


at  the  instant  of  explosion,  and,  as  Mr.  Clerk  says,  are 
merely  averages ;  and  it  may  be  taken,  that  coal  gas  mix- 
tures with  air  give  upon  explosion  temperatures  ranging 
from  800°  C.  (1500°  F.  nearly)  to  nearly  2000°  C.  (3600° 
F.),  depending  upon  the  dilution  of  the  mixture.  Since 
cast  iron  will  melt  when  subjected  to  a  prolonged  heat  oi 
about  2000°  F.  (p.  89),  the  heat  of  explosion  would  de- 
stroy the  working  surface  if  it  were  not  cooled  by  some 
artificial  means;  but  with  the  means  employed,  cylinders 
have  been  used  for  years,  and  a  wearing  surface  main- 
tained as  perfect  as  in  the  steam-engine.  The  glow  result- 


FIG.  65. 

ing  from  the  explosion  has  been  observed  by  inserting  in 
the  cylinder  a  small  tube  containing  a  strong  glass  through 
which  one  could  look. 

13O.  Theory.  We  will  consider  the  bi-re volution 
compression  system.  Fig.  65  is  an  actual  indicator  diagram 
taken  from  a  10  horse-power  Otto  engine  during  an  experi- 
ment in  the  Institute,  except  that  we  have  added  the  part 
A\  C  B  to  represent  the  combustion  chamber,  and  reduced 
the  linear  dimensions  one  half.  It  is  a  fair  sample  of  many 
others  that  were  taken.  A  D  is  the  atmospheric  line,  1  D 
the  stroke  of  the  piston,  A  1  the  clearance,  2  5  the  com- 
pression line,  5  C  the  explosion  line,  C  7  the  expansion  line. 
The  explosion  is  nearly,  but  not  quite,  instantaneous,  as 


254  HEAT   ENGINES.  [l30.J 

shown  by  the  line  5  C — the  piston  moving  n  very  short 
distance  before  the  explosion  is  complete.  During  the  tak- 
ing in  of  the  charge  the  pressure  follows  the  line  12; 
during  compression,  the  line  25  ;  during  explosion,  the  line 
5  C\  during  expansion,  the  line  C  7;  during  exhaust,  78; 
during  the  fourth  stroke  the  products  of  combustion  are 
forced  out,  following  81. 

In  subjecting  these  operations  to  analysis,  we   proceed, 
as  before,  to  construct  an  ideal  indicator  diagram,  in  which 
we  assume  that  the  explosion  takes  place    instantaneously 
at  the  remote  dead  point — that  the  expansion  and  compres- 
sion  lines  are  adiabatic — that  the  fall  of 
pressure  at  the  end  of  the  stroke  takes 
place  without   change   of   volume — and 
that  the  change  is  taken  in  and  expelled 
at  atmospheric  pressure  ;  thus  producing 
a  diagram  like  0,  1,2,  3,  Fig.  66,  in  which 
C  O  represents  the  pressure  of  the  atrnos- 


FIG.  66.  phere. 

Letting  the  subscripts  corresponding  to 

the  corners  of  the  diagram  represent  the  corresponding 
states ;  then  will  the  heat  energy  developed  by  the  explosion 
be  the  area  3012  indefinitely  extended  to  the  right  between 
the  adiabatics  03  and  12  prolonged;  the  value  of  which 
for  each  pound  of  the  substance  will  be 

77,  -  Cv  (rt  -  r0), 

as  in  Exercise  3,  page  64.  The  heitf  rejected  in  passing 
from  state  2  to  state  3  will  be,  similarly, 

II,  =  ^v(r,-r3); 
hence  the  efficiency  will  be 

Tl~l°2  (T-    ~  ^  =  l~  I*  -  r'-  (245) 

Since  #„  =  vt  and  vt  =  #„  we  have  from  equation  (42),  if  y 


[130.  J  THEORY.  255 

be  constant,  and  the  same  for  the  curve  of   expansion  as 
for  compression, 

*"„        r. 


!  5          (246) 

according  to  which  it  appears  that  the  efficiency  depends 
only  upon  the  ratio  of  the  temperatures  just  before  and  just 
after  compression — or,  generally,  upon  the  temperatures  at 
the  extremities  of  either  adiabatic,  but  otherwise  is  inde- 
pendent of  the  temperature  of  the  explosion. 
The  work  per  pound  will  be 

To  find  this  work  in  terms  of  p  and  v,  we  have  pv  =  I?T&$ 
in  equation  (2),  and  J2  =  (y  —  1)  (7V,  as  in  the  answer  to  Ex- 
ercise 7,  page  59  ;  .  ' .  Cv  r0  =  ^°  v°  ,  as  at  the  top  of  page 
65,  and  similarly  for  p^  -y,,  r^  &c. ;  hence, 

ffib.  =  ^3  (P,  v*  -  A  <  -  P.  «,  +pt  <0.         (248) 

This  may  be  further  reduced  by  means  of  equation  (42), 
and  making  U^  =  L\  -r-  vs  =  the  work  done  per  cubic 
foot  of  the  mixture,  r  =  vt  -f-  «„  the  ratio  of  expansion, 
and^?j  =  p9  ?'y,  we  have 

grfi  =P>-P»      ^Y~1  ~  1  .  (249) 

The  mean  effective  pressure  on  a  square  foot  of  the  pis- 
ton, or  the  energy  developed  per  foot  of  volume,  distrib- 
uted over  four  strokes,  will  be 

(250) 


256  HEAT   ENGINES. 

Let  N  be  the  number  of  revolutions  per  minute  ;  S, 
the  average  piston  speed  ;  I,  the  length  of  stroke  ;  ffP,  the 
horse-power;  TF,  the  work  in  foot-pounds  developed  by 
the  piston  per  minute  ;  w,  the  pounds  of  the  mixture  at 
each  explosion  ;  A,  the  area  of  the  piston  ;  then^ 

8=  *Nl; 
W  =  33000  HP  =  ZpJAN  (251) 


Work  done  per  pound  per  single  stroke  will  be 
W 


Volume  swept  over  by  the  piston  per  stroke 

Al  =  m^ffP-  (253) 

Volume  swept  over  by  tlie  piston  per  pound  of  the  mixture 
per  stroke 

«.-«,=    *L.  (254) 

4^e 

131.  The  furnace.  The  smaller  the  combustion 
chamber,  compared  with  the  volume  swept  over  by  the  pis- 
ton, the  more  efficient  will  be  the  charge,  as  shown  by  equa- 
tion (246),  since  TS  -f-  r0  will  be  smaller  the  smaller 
v0  -r-  v3  as  shown  by  equation  (42).  But,  on  the  other 
hand,  the  smaller  v0  =  vl  is  when  the  explosion  takes  place, 
the  greater  will  be  pl  and  rl  after  the  explosion,  as  shown 
by  equation  (2)  ;  but  as  the  temperature  of  explosion  is 
very  high  in  practice,  there  will  be  a  practical  inferior  limit 
to  the  size  of  the  combustion  chamber,  which  must  be  deter- 
mined by  a  protracted  use  of  the  engine.  In  the  "  Otto,'' 
with  which  experiments  were  made  at  the  Institute,  the 
combustion  chamber  was  0.38  of  the  entire  volume  of  the 
cylinder,  and  about  the  same  relation  exists  in  some  other 
engines. 


[132.]  THEORETICAL   ENERGY   OF   THE   GAS,  257 

The  efficiency  of  the  explosion,  or,  as  we  may  say,  the  effi- 
ciency of  the  furnace,  is  not  perfect.  Experiments  made 
by  Mr.  D.  Clerk  in  closed  vessels  of  fixed  volume,  on  the 
supposition  that  the  absolute  temperature  varied  as  the 
absolute  pressure  at  constant  volume,  found  that  the  heat 
evolved  varied  from  50  to  60  per  cent  of  the  theoretical 
(The  Gas  Engine,  p.  182)  ;  the  latter  being  nearly  the 
highest  value  found  in  any  case,  while  in  many  cases  it  is 
considerably  less  than  the  former.  Thus,  for  a  mixture  of 
air  and  Oldham  gas,  he  found 

Fraction  of  gas,  vol.  ^,     TV,     ^     TT»       i>        T 
Heating  efficiency     0.40,  0.48,  0.50,  0.46,  0.40,  0.37. 

This  shows  that  the  furnace  efficiency  diminishes  with 
the  richness  of  the  gas  when  the  gas  exceeds  y1^-  of  the  vol- 
ume of  the  mixture  (ibid,.,  p.  113).  The  table  in  Article  129 
shows  that  the  efficiency  in  that  case  varied  from  a  little 
below  to  a  little  above  50  per  cent,  when  the  initial  tempera- 
ture was  17°  C.  The  initial  temperature  in  the  engine 
will  be  considerably  above  this,  which,  added  to  the  facts 
that — the  pressure  in  the  cylinder  may  be  less  than  that  of 
the  atmosphere — a  part  of  the  products  of  combustion  will 
be  retained — possible  leakage — and  imperfect  action — make 
it  advisable,  in  the  absence  of  actual  measurements,  to  con- 
sider the  efficiency  of  the  explosion  as  not  more  than  0.45 
of  that  indicated  by  the  chemical  composition.  The  cause 
of  the  large  difference  between  the  theoretical  and  actual 
heat  developed  is  not  well  known.  It  is  found  that,  gen- 
erally, the  best  results  are  obtained  when  the  volume  of 
air  is  6  or  7  times  that  of  the  gas,  so  that  the  volume  of 
the  gas  for  each  charge  will  be  |  or  £  of  the  volume 
swept  over  by  the  piston  in  one  stroke. 

132.  To  find  the  work  and  efficiency  in 
terms  of  the  theoretical  energy  of  the  gas. 
Let  &0  be  the  energy  of  the  gas  in  thermal  units,  devel- 


258  HEAT    ENGINES.  [133.J 

oped  by  the  explosion  of  one  pound  of  the  gas  as  deter- 
mined from  its  chemical  composition,  J\c  the  dynamic 
equivalent,  m  the  coefficient  of  reduction  of  its  efficiency, 
J,,  r,,  P,,  />„  the  temperatures  and  pressures,  respectively, 
which  would  result  at  the  states  1  and  2,  if  the  efficiency, 
m,  were  unity,  and  the  expansion  adiabatic  ;  and  assuming 
that  the  entire  energy  of  the  explosion  is  communicated  to 
the  mixture,  we  have 

JTe=Jt,=  C,(r,-r.);  (255) 


and  establishing  an  equation  in  the  same  manner  as  (247)  we 
have,  for  the  indicated  work  per  pound  of  the  mixture  per 
stroke  of  the  explosion,  or  per  bi-revolution  of  the  engine, 


(257) 
hence,  the  indicated  efficiency  of  the  plant  will  be 

(258) 


133    The  expansion  and  compression,  curves. 

Since  there  is  a  flow  of  heat  from  the  working  fluid  to  the 
water  jacket  and  the  reverse,  the  curves  will  not  be  strictly 
adiabatic,  neither  will  they  be  isothermal.  Their  character 
may  be  more  accurately  determined  from  a  study  of  an 
actual  diagram.  Assuming  that  they  follow  the  law 

p  v1  =  constant^ 

Messrs.  Brooks  and  Stewart  found,  for  the  curve  between 
6  and  7,  Fig.  65,  x  =  1.363  ;  and  for  the  compression  curve, 


|134.]  EXPERIMENTAL    RESULTS.  259 

x  =  1.335.  Professors  Ayrton  and  Perry,  by  an  experi- 
ment which  they  confess  was  not  as  accurate  as  the  above, 
found  for  the  expansion  curve,  x  =  1.479,  and  for  the  com- 
pression curve,  x  =  1.304.*  These  results  show  that  it  is 
much  more  nearly  adiabatic  than  isothermal.  The  value  of 
y  used  in  our  analysis  should  be  less  than  1.4,  the  value  for 
air,  because  the  presence  of  the  hydrocarbon  of  the  mix- 
ture will  reduce  the  ratio  of  the  specific  heats  ;  but  since 
the  quantity  of  air  predominates,  we  may,  in  the  absence  of 
actual  measurements,  use  y  =  1.4.  Air  behaves  so  nearly 
like  a  perfect  gas  that  this  value  would  be  practically  con- 
stant, even  for  the  highest  temperatures,  if  the  expansion 
were  adiabatic. 

134.  Experimental  results.     Messrs.  Brooks  and 

Steward,  during  the  summer  of  1883,  made  a  thorough  test 
of  an  Otto  engine,f  from  which  we  make  the  following 
abstract.  Dimensions  of  the  cylinder,  8£  inches  diameter,  14 
inches  stroke.  The  air  and  gas  used  in  mixtures  were  both 
measured  by  a  gas  metre,  and  it  was  found  that  when  the 
volume  of  air  used  was  7.1  times  that  of  the  gas,  the  best 
indicated  results  were  obtained.  The  diagram  taken  during 
the  19th  test  is  shown  in  Fig.  65,  with  the  linear  dimensions 
reduced  to  one  half  their  original  value.  During  this  test 
it  was  found  that : 

Vol.  air         a  ?0  Weight  of  air        -.  0  #Q 

^^ =  6.63  ;  T_  .y       •' =  13.68. 

Vol.  gas  Weight  oj  gas 

About  one  haL  the  heat  of  explosion  was  carried  away  by 
the  water  jacket.  The  temperatures  were  computed  by 
means  of  equation  (2),  the  volumes  and  pressures  being 


*  Phil.  Mag.,  1884,  (2),  65. 

f  Graduation  Tliesis  at  Stevens  Institute  of  Technology,  1883  ;    Van 
Nostrand's  Engineering  Magazine,  1884,  Feb.,  pp.  90-104. 


260  HEAT   ENGINES.  [134.] 

measured  from  the  indicator  diagram.     The  specific  heats 
of  the  mixture  were  computed  to  be 

cp  =  0.268  ;     cv  =  0.196  ;     .  • .  y  =  1.37. 

The  complete  combustion  of  the  gas,  determined  from  its 
chemical  composition,  gave  9070  calories  per  kilog.,  or 
617.5  B.  T.  U.  per  cubic  foot.  From  23.5  to  25.6  cubic 
feet  of  this  gas  were  used  per  indicated  horse-power. 

The  mean  effective  pressure  was  about  58  pounds  per 
square  inch  per  stroke  of  the  explosion ;  and  if  there  was 
an  explosion  for  every  fourth  stroke  the  average  pressure 
was  Ity  Ibs.  for  every  stroke.  The  average  number  of 
revolutions  was  nearly  155  per  minute. 

Temperature  of  the  exhaust  gases,  from  720°  F.  to  778°  F., 
as  determined  by  a  pyrometer. 

The  indicated  efficiency  was  18  per  cent  of  the  total  heat 
of  combustion  of  the  gas,  rnd  the  effectual  efficiency  for  the 
plant,  as  determined  by  a  brake,  was  14^  per  cent. 

The  experiments  upon  the  "  Otto,"  of  various  sizes  and  in 
distant  parts  of  the  world,  have  been  numerous,  giving 
efficiencies  of  15,  16,  17,  and  18  per  cent. 

From  these  results  it  will  be  seen  that  the  explosive  gas- 
engine  gives  the  highest  indicated  efficiency  for  the  plant 
of  any  system  thus  far  considered.  For  intermittent  work 
and  cost  of  attendance,  it  has  an  advantage  over  the  steam- 
engine.  On  the  other  hand,  the  engines  are  much  larger  for 
the  same  power,  and  are  thus  objectionable  for  very  large 
powers,  and  the  cost  of  fuel  for  steady  running  is  greater, 
since  a  heat  unit  in  the  form  of  gas  costs  considerably  more 
in  the  market  than  a  heat  unit  in  the  form  of  coal.  Cir- 
cumstances must  decide  what  class  of  engines  is  most  econ- 
nomical  for  a  particular  case. 

The  following  is  an  analysis  by  Professor  T.  B.  Stillman,  of  Stevens 
Institute,  of  the  gas  used  by  Messrs.  Brooks  and  Stewart.  The  gas 
was  taken  from  the  mains  supplied  by  the  Hoboken  Gas  Company  : 


[134.]  EXPERIMENTAL   RESULTS.  261 

By  volume. 

H Hydrogen 395 

CH4 Marsh  gas 373 

N Nitrogen 082 

C3H6,  Average. . .    .Heavy  hydrocarbons 066 

CO  Carbonic  oxide 043 

O Oxygen 014 

HjOj.COj.HiiS,  &c.. Impurities,  &c 027 

1.000 
By  weight  its  composition  is  found  to  be  : 

Cu.              Densi-          Kilos,  per  Wt  p. 

metres.          ties.*               en.  m.  unit. 

H                  .395     X       .087     =       .035  .058 

CH4              .373     X       .694     =       .258  .426 

N                  .082     X     1.215     =       .099  .163 

C3H6,  Av'e  .066     X     1.84      =      .121  .200 

CO                .043     X     1.215     =       .052  .086 

O                  .014     X     1.388     -       .019  .031 

H3O,,  &c.    .027     X     ~.8        =       .022  .036 

1.000     X       .606     -      .606       1.000 

HEATING  POWER  OF  THE  GAS. 

Upon  complete  combustion  the  gas  develops  heat  per  cubic  metre,  as 
follows : 

Calories.  Calories. 

fromH  29060  X  .035  =  1020 

"     CH4  11710  X  .258  =  3020 

"     C3H6,&c.  11000  X  .121  =  1330 

"      CO  2400  X  .052  =       125 

per  cu.  m.  5495c. 
and  per  kilog.  gas  —         =  9070  calories. 

Expressed  in  British  measures,  one  cubic  foot  of  gas  develops  617.5 
heat  units. 

AlR  NECESSARY  FOR  COMPLETE   COMBUSTION  AND  THE  PRODUCTS  OF 
COMBUSTION. 

In  order  to  determine  the  amount  of  air  to  be  supplied  for  complete 

*  Scho'ttler:  Die  Gasmaschinc,  p.  77.  By  "density"  is  meant  the 
weight  of  one  cubic  metre  in  kilogrammes.  As  will  be  seen  from  the 
above,  one  cubic  metre  of  the  gas  in  question  weighs  0.606  kilos. 


262  HEAT   ENGINES.  [134.] 

combustion,  it  is  necessary  to  ascertain  the  quantity  of  oxygen  that  is 
taken  into  chemical  combination  by  the  several  combustible  constituents 
of  the  gas. 

2H  +  O    =  H,O 
by  volume     2+1=2 
by  weight     2  +  16  =  18 

CH4  +  40  =.  C0a  +  2H,O 
by  volume    2  -j-    4   =  2       +4 
by  weight    16  -j-  64   =  44     +36 

C,H.  +  90  =  3COa  +  3HaO 
by  volume    2  +  9      =6       +6 
by  weight  42  +  144  =  132   +54 

CO  +  O     =  CO, 
by  volume    2  +  1      =2 
by  weight    28+16  =44 

The  combining  proportions  per  unit  of  the  several  constituents  is  : 

By  volume— 

1H       +  iO    =  !HaO 
1CH4  +  20    =  ICO,  +  2HaO 
1CSH.  +  4*O  =  SCO,  +  3HaO 
ICO     +  |O    =  ICO, 

By  weight— 

1H       +80    =  9H20 

1CH4  +  4O  =  VCO,  +  JH,0 
1C.H.  +  ^O  =  ^CO,  +  ?HaO 
ICO  +  K>  =  V-CO, 

The  volume  of  oxygen  required  for  the  combustion  of  1  volume  of 
gas  is : 

H^   .395  X    i  =  .197 

CH«    .373  X    2  =  .746 

C,H«  .066  X  4i  =  .297 

CO    .043  X    4  =  .022 

1.262 
less  O  in  gas  .014 014 

1.248 

Taking  oxygen  as  21  per  cent  in  atmospheric  air,  the  volume  of  air  re- 
quired is 

1-248 
— — —  =  5.94  per  volume  gas, 

or  the  entire  volume  is  6.94  times  the  volume  of  gas. 


[134.]  EXPERIMENTAL   RESULTS.  263 

Since  air  weighs  1.251  kilos,  per  cu.  metre,  the  ratio  by  weight  is 

' 


From  the  combustion  of  1  unit  weight  of  gas  with  12.26  air  there  re- 
sults 13.26  units  weight  of  a  mixture  the  composition  of  which  will  be  : 


S(CH4) 426  X  V  =  1.171  ) 
(C3H6) 200  X  V-  =  .629  f 
(CO) 086  X  -V-  =  -135  ) 

•i 


1.93 


(H)  ................  058  X    9=    .522) 

H80-<   (CH4)  ..............  426  X    J  =    .958V  1.74 

(C3H6)  .............  200  X    ¥  =    .257) 

N  j  from  the  air...  ........  ,  ......  9.407)  q  .. 

*  \  in  gas  itself  .....................  163  f 

Impurities  in  gas  ......................................  0.03 


13.27 
Per  unit  weight  of  mixture  the  composition  will  be  : 

CO,  ..........................................  146 

HaO  ............................................  131 

N  ..............................................  721 

Impurities  ...............................  ......  002 

1.000 

The  volume  which  13.27  kilos,  of  products  of  combustion  will  occupy 
is  found  from  the  known  volumes  of  the  constituent  gases  as  follows  : 

en.  m.  per 

kilos.       kilo.        cu.  m. 

CO2    1.93  X  -524  =  1.011 

HS0    1.74  X  1,28  -  2.227 

N       9.57  X  .823  =  7.876 

Impurities    .03  X  -.9    =    .027 

11.141 

The  products  of  combustion  then  occupy  11.141  cu.  m.  to  every  kilog. 
of  gas.  To  find  the  ratio  per  cu.  metre  of  gas  we  have  simply  to  multi- 
ply by  .606  the  number  of  kilos,  in  a  cubic  metre,  and  we  get  6.751  as 
the  result.  As  there  is  necessary  6.94  cu.  m.  of  mixture  of  air  and  gas 
to  every  cu.  m.  gas,  it  is  seen  that  by  combustion  a  contraction  of  2.7 
per  cent  takes  place. 

When  there  is  an  excess  of  air  present,  as  is  always  the  case  in  prac- 
tice, the  contraction  becomes  less  in  proportion,  and  may  be  considered 
to  be  about  2  per  cent.  In  the  following  thermodynamic  computations 
no  account  is  taken  of  this  contraction. 


264  HEAT   ENGINES.  [134.] 

SPECIFIC  HEATS  AND  THEIR  RATIO. 

The  specific  heats  of  the  products  of  combustion  are  determined  from 
the  specific  heats  of  the  several  component  gases  as  follows : 
Specific  heat  at  constant  pressure  (water  =  1). 

{  .2169  X  .146  (CO,) 

„    _      I  .4805  X  -131  (H2O)  =  .uozy 

Cp  -    1  .2438  X  .721  (N)  =  .1758 

[  ~A  X  .002  (impurities)  =  .0008 

Specific  heat  at  constant  volume  (water  1). 

f  .1714  X  .146  (CO,)  =  .02501 

r    _  .3694  X  .131  (H20)  =  .0484  I    1QRS 

Cv  ~  .1727  X  .721  (N)  =  .1245  f -iyt 

[      -.3  X  .002  (impurities)  =  .0006  J 

The  ratio  of  these  specific  heats  is  the  exponent  of  adiabatic  expansion, 
and  is  found  to  be  : 


Since  there  is  always  an  excess  of  air  present,  these  values  will  be 
somewhat  modified  by  that  fact.  From  the  metre  records  of  test  19  the 
ratio  of  air  to  gas  by  volume  was  found  to  be  6.63  to  1  ;  by  weight  the 
ratio  is 

6.63  X  1.251 
1  X  .606 

Since  for  complete  combustion  only  12.26  parts  of  air  by  weight  are 
needed,  there  are  1.42  parts  in  excess.  The  specific  heats  of  air  being 
(7P  =  .2375  and  Cv  =  .1684.  the  effect  of  the  excess  of  air  will  be  to 
reduce  the  specific  heat  slightly. 

(.2712  X  13.26)  4-  (.2375  X  1.42) 
Cp  =  14.68 

„    _  (.1985  X  13.26)  +  (.1684  X  1.42) 

14.68 

CP          .268 
7  =  T\    =  196   =  °7 


EXERCISES. 

1.  Required  the  horse-power  and  efficiency  of  a  gas-en- 
gine whose  cylinder  is  8£  inches  diameter,  stroke  14: 
inches,  revolutions  160  per  minute,  charge  every  fourth 
stroke,  combustion  chamber  0.38  of  the  volume  of  the  cylin- 
der, heat  of  combustion  of  one  pound  of  the  gas  1G32>> 


[134.]  EXPERIMENTAL   RESULTS.  265 

B.  T.  IL,  volume  of  working  air  7  times  that  of  the  gas, 
weight  of  gas  say  -^  of  that  of  the  air,  efficiency  of  the 
furnace  0.60  of  the  theoretical,  y  =  1.38.  (These  are 
approximately  the  conditions  of  the  engine  and  test  in 
Brooks  and  Stewart's  experiments.) 

We  find- 
Area  of  the  piston,  sq.  in =  56.75. 

Piston  displacement  per  stroke,  cu.  ft. =    0.46. 

Yol.  of  air  taken  in  each  fourth  stroke,  £  of  0.46  =  0.402. 

Pounds  of  air  for  each  charge 0.402  X  0.08  =  0.032. 

Pounds  of  gas  for  80  charges..  .Ty  X  0.032  X  80  =.  0.150. 
Work  per  Ib.  gas,  eq.  (257),  ft.-lbs 0.60  X  778 

X  16326  [1  -  0.380-38] =  2344970. 

IHP.  for  the  0.15  Ibs.  gas   2344970  X  0.15 =    10A 

Indicated  efficiency,  (258) E  =  0.185. 

If  m  =  0.55  we  would  have  : — 

Indicated  HP.  for  0.15  Ib.  gas =       9.7. 

Indicated  efficiency E=  0. 169. 

These  last  results  agree  very  nearly  with  the  measured  re- 
sults of  Brooks  and  Stewart — the  horse-power  being  the 
same  and  the  efficiency  about  one  per  cent  less  than  their 
best  result. 

2.  Required  the  horse-power,  efficiency,  pounds  of  air  and 
of  gas  per  minute,  of  a  gas-engine  having  a  10-inch  cylin- 
der, 16-inch  stroke,  making  150  revolutions  per  minute, 
charge  every  fourth  stroke,  combustion  chamber  0.4  of  the 
volume  of  the  whole  cylinder,  heat  of  combustion  of  one 
pound  of  the  gas  18000  B.  T.  U.,  volume  of  working  air 
6$  times  that  of  the  gas,  weight  of  the  gas  0.55  that  of  an 
equal  volume  of  air,  12J  cubic  feet  of  air  to  weigh  a  pound, 
efficiency  of  furnace,  0.50,  and  y  =  1.4. 


HEAT  ENGINES.  [135.J 


THE  PETROLEUM   ENGINE. 

135.  Naphtha  Engine.  In  petroleum  engines,  the 
working  fluid  is  either  petroleum,  or  some  of  its  products. 
Bray  ton's  petroleum  engine  was  a  modification  of  that  in- 
ventor's gas-engine.  The  products  of  combustion  entered 
the  cylinder,  and  in  this  respect  was  similar  to  the  gas- 
engine  ;  but  combustion  was  gradual,  the  indicator  diagram 
showing  that  it  was  made  at  nearly  constant  pressure  up  to 
the  point  of  cut-off.  Mr.  Clerk  concludes  from  his  ex- 
periments with  a  5  HP.  engine  that  it  utilizes  about  6  per 
cent  of  the  heat  of  the  petroleum.  Professor  Thurston, 
from  an  experiment  with  one  of  these  engines,  concluded 
that  32.06  cubic  feet  of  gas  were  consumed  per  IHP. ; 
but  Mr.  Clerk  concludes  that  the  same  experiment  shows  a 
consumption  of  55.2  cu.  ft.  per  IHP.  per  hour.  (The 
Gas  Engine,  p.  158.) 

"We  will  consider  only  a  recent  form,  called  the  Naphtha 
Engine,  of  which  Fig.  67  is  an  external  view.  In  this  en- 
gine the  products  of  combustion  do  not  enter  the  cylinder, 
but  the  same  substance  is  used  for  the  fuel  and  working 
fluid.  A  small  plunger  pump  near  D,  but  not  shown  in  the 
figure,  worked  by  the  engine,  forces  some  liquid  naphtha 
into  the  boiler  F  at  each  revolution,  a  part  of  which  is 
conducted  from  the  boiler  down  the  tube  H  to  the  coiiil  ms- 
tion  chamber,  or  furnace,  below  the  valve  chest  A,  and  is 
there  burned.  At  E'\&  an  opening  into  the  tube  II  through 
which  air  may  be  forced  to  increase  the  rate  of  combustion. 
The  heat  of  the  burning  naphtha  vaporizes  that  remaining 
in  the  boiler,  and  the  vapor  thus  generated  is  used  in  the 
engine  in  precisely  the  same  manner  as  if  it  were  steam ; 
and  the  law  of  its  action  in  the  engine  is  precisely  the 
same  as  steam,  as  may  be  inferred  from  Fig.  68,  which  is  a 
copy  of  an  indicator  diagram  taken  by  Doty  and  Beyer, 


[136.; 


EXPERIMENTS. 


267 


FIG.  67. 


in  the  experiments  referred  to  be- 
low. The  ratio  of  expansion,  as  here 
shown,  is  about  2,  but  it  may  be 
varied  at  pleasure.  The  drop,  when 
the  exhaust  opens  at  the  end  of  the 
stroke,  is  sudden,  and  the  back  pres- 
sure and  compression  lines  are  good. 
The  depression  of  the  steam  line, 
showing  initial  expansion,  is  prob- 
ably due  to  the  setting  of  the  valve, 
since  the  diagram  from  one  of  the 
cylinders  was  free  from  this  de- 
fect. 

If  the  diagram  be  freed  of  its  irregularities  and  of 
compression,  it  would  be  analyzed  precisely  like  the  steam- 
engine,  and  the  solution  in  Articles  110,  111,  and  112  would 
be  applicable.  Probably  Article  110  represents  the  case 
more  nearly,  since  the  cylinders  are 
very  near  the  boiler,  and  receive  heat 
continually  from  it.  But  the  analysis 
cannot  be  carried  out  numerically,  since 
the  physical  properties  of  Naphtha  are 
not  sufficiently  well  known.  The  latent 
heat  of  evaporation  at  varying  pressures  is  not  known, 
nor  the  value  of  R  in  the  equation  p  v  =  R  r — if 
indeed  it  is  constant  for  the  vapor.  We  will,  however, 
after  giving  the  results  of  some  experiments,  make  an  ap- 
proximate solution. 

136.  Experiments.  Messrs.  Doty  and  Beyer  made 
experiments  upon  a  naphtha  engine,  of  which  the  following 
is  a  summary  of  their  report  :* 

Three  single-acting   trunk  engines  were  connected  to  the 


FIG.  68. 


*  Graduation  Thesis  of  Paul  Doty  and  Richard  Beyer,  Stevens  Institute 
of  Technology,  Hoboken,  1888.     The  Iron  Age,  July,  188a 


268  HEAT   ENGINES.  [136.] 

crank  shaft,  the  cranks  making  successive  angles  of  120° 
with  each  other.  The  lower  ends  of  the  coils  constituting 
the  boiler  were  connected  with  the  pump,  and  the  upper 
ends  entered  a  common  chamber. 

Diameter  of  cylinders,  each 3£  inches. 

Stroke  of  pistons,  each 4£ 

Piston  displacement,  each 37£  cu.   in. 

Admission  ports,  each fa  x  2-jV  inches. 

Exhaust  ports,  each -fa  x  2-iV 

Diameter  of  pump 1^      " 

Stroke  of  pump If      " 

Travel  of  main  valves,  each f      " 

Clearance  of  cylinders,  each. .  f 1.86  cu.  in. 

or  5J  per  cent  of  piston  displacement. 
Boiler,  seven  spirals  of  four  coils  each. 

Coils,  copper  tubing  outside  diam £      " 

Height  and  diameter  of  coils,  each 12      " 

Burner  had  26  openings,  diam.  of  each W      " 

Heating  surface 12  sq.  ft. 

RESULTS. 

Average  revolutions  per  minute,  number 280.7. 

Total  Indicated  HP.  from  the  three  cylinders 2.81. 

Mean  effective  pressure,  Ibs.  per  sq.  in. ,  about 35. 

Naphtha  burned  per  IHP.  per  hour,  Ibs 3.53. 

Price  of  naphtha,  June  5th,  cents  per  gallon 10. 

Cost  per  IHP.  per  hour,  cents 6.2. 

Heating  surface,  square  feet  per  IHP 4.3. 

Water  used  to  condense  the  exhaust  naphtha,  Ibs.  per  IHP.    per 

hour :• 25594. 

Increase  of  temperature  of  condensing  water,  degrees  F 3.9. 

Naphtha  passing  through  condenser  per  hour,  Ibs 421. 

Temperature  of  the  stack,  degrees  F.,  about 685. 

Specific  gravity  of  the  naphtha,  that  of  water  being  unity 0.683. 

One  gallon  weighed,  pounds . .  5.69. 

Assuming  that  the  vapor  was  saturated,  three  elements  of 
this  data  give,  for 

2559-i  X  3  9 
the  latent  heat  of  evaporation  -         1— - — -        =  237 

thermal  units,  at  atmospheric  pressure. 


[136.]  EXPERIMENTS.  269 

In  order  to  determine  the  relation  between  the  tempera- 
ture and  pressure  of  saturated  vapor,  these  experimenters 
devised  a  special  thermometer  with  which  they  determined 
the  temperature  of  the  vapor  in  the  steam  chest,  and  at  the 
same  time  determined  the  pressure  by  means  of  a  pressure 
gauge.  These  measurements  gave 

for  35  Ibs.  gauge  pressure  a  temp,  of  225°  F.  ; 

"   47    "       "  "        "     "        "  242°  F.  ; 

"   60    "       "  "        "     "        "  258°  F. 

Substituting  in  equation  (80),  page  97,  these  values  of  p 
reduced  to  their  equivalent  in  pounds  per  square  foot,  and 
the  corresponding  temperatures  reduced  to  the  absolute 
scale,  and  finding  the  values  of  A.,  J3,  C,  we  have 


**.,=  8.4818  -_.;  (259) 

in  which 

log^B  =  2.949092  ;         log^C  =  5.796469. 

Making^  =  2116.2  this  formula  gives  r  =  602.62°  or  T 
=  141.96°  F.  Naphtha  has  not  a  fixed  boiling  point.  In 
an  experiment  it  began  to  boil  at  60°  C.,  and  as  the  more 
volatile  parts  passed  off,  the  temperature  gradually  in- 
creased, and,  in  the  course  of  twenty  minutes,  it  raised  to 
68°  C.,  giving  a  mean  of  64°  C.  =  147°  F.  The  value 
found  by  the  formula  agrees,  approximately,  with  the  lower 
observed  temperature,  60°  C.  =  140°  F. 

To  find  the  volume  of  one  pound  of  the  saturated  vapor 
of  naphtha  at  atmospheric  pressure,  we  have,  from  equation 
(84),  page  98, 

«.  =  0-0234  +.  -8  =  7.69  cu.  ft., 


in  which  0.0234  is  the  value  assigned   for   one  pound   of 
liquid  naphtha  at  60°  F. 


270  HEAT   ENGINES.  [137.] 

It  was  found  that  the  naphtha,  exposed  to  an  atmospheric 
pressure  of  29.982  inches,  at  a  temperature  of  70°  F., 
evaporated  at  the  rate  of  0.092094  Ibs.  per  square  foot  per 
hour. 

Some  of  the  values  found  by  these  experimenters  differ 
largely  from  those  given  by  others.  Thus,  the  boiling  point, 
as  found  by  them,  is  between  116°  F.  and  167°  F.,  but  Box 
On  Heat,  page  14,  gives  306°  F.  on  the  authority  of  Ure. 
The  latter  must  be  an  error.  They  found  the  latent  heat  of 
evaporation  to  be  237 ;  while  Box,  page  18,  gives  184  on 
the  authority  of  Ure.  If  lire's  figures  were  reversed,  giving 
184°  F.  for  the  boiling  point  and  306  for  the  latent  heat  of 
vaporization,  they  would  appear  more  rational ;  still  we  can- 
not say  what  is  correct.  They  give  0.683  for  the  specific 
gravity  at  60°  F.,  while  Rankine,  in  Table  II.  of  the  Steam 
Engine,  gives  0.848  at  32°  F.  ;  and  both  cannot  be  correct. 
The  discrepancies  may  be  due  in  part  to  the  difference  in 
the  chemical  composition  of  the  different  specimens.  The 
relation  between  the  pressure  and  volume,  given  in  equation 
(259),  is  considered  only  approximate,  not  only  on  account 
of  the  heterogeneous  character  of  the  substance,  but  also  be- 
cause it  depends  upon  three  experiments  only,  whereas  there 
should  be  a  larger  range  of  experiments  in  order  to  test  its 
accuracy  and  reliability.  The  results  will,  however,  be  sub- 
jects for  comparison  for  future  experimenters. 

137.  Efficiency  of  the  Naphtha  Engine.     In 

the  absence  of  a  determination  of  the  calorific  power  of  the 
naphtha  used  in  the  above  experiment,  we  will  assume  that 
the  heat  of  combustion  is  22000  thermal  units  per  pound, 
since  its  value  would  be  22274  if  the  composition  were 
C.  HM. 

For, 

6  X  12  X  14544  =  1047168 
14  X  1  X  62032  = 


)1915616(  22274. 


[138.]  EFFICIENCY   OF   FLUID.  271 

There  was  consumed  3.53  pounds  of  naphtha  per  IHP. 
per  hour,  hence  the  indicated  efficiency  of  the  plant,  includ- 
ing fuel,  furnace,  and  engine,  was 

T7  33000  X  60 

*  ~ 


3.53  X  22000  X  778 

or  nearly  3£  per  cent.  It  is  a  good  3  horse-power  en- 
gine, that—  including  fuel,  furnace,  boiler  and  engine— 
yields  this  efficiency.  If  the  particular  naphtha  used  were 
richer  in  hydrogen  than  that  assumed,  or  rather  if  its  chemi- 
cal composition  gave  23000  thermal  units,  the  efficiency 
would  be  reduced  to  3.1  per  cent. 

The  cost  of  running,  6.2  cents  per  IHP.  per  hour,  is 
not  a  measure  of  the  efficiency,  but  of  the  economy.  A 
steam-engine,  run  by  the  waste  fuel  of  a  saw-mill,  may  cost 
nothing  *for  fuel  ;  while  the  same  engine  run  with  anthracite 
coal  may  cost  many  dollars  daily  for  this  item,  while  as  a 
heat  engine  the  efficiency  should  be  the  same  in  the  two 


138.  Efficiency  of  fluid.  Any  solution  of  this  part 
of  the  problem  will  necessarily  be  approximate,  since  some 
of  the  data  must  be  assumed ;  and  yet  such  a  solution  may 
give  some  idea  of  its  probable  efficiency,  and  hence  of  the 
efficiency  of  the  furnace  and  boiler.  Regnault  found  the 
specific  heat  of  petroleum  to  be  0.434,  and  we  will  assume 
it  to  be  the  same  for  liquid  naphtha.  The  latent  heat  of 
evaporation  at  atmospheric  pressure  is  237  B.  T.  U.  per 
pound,  as  found  above ;  but  the  law  of  change  with  tem- 
perature and  pressure  is  not  known.  As  this  value  approxi- 
mates more  nearly  to  that  of  acetic  acid  than  any  other  sub- 
stance now  before  us  (see  Article  74  of  Addenda),  we 
will  assume,  although  otherwise  quite  arbitrary,  that 

he  =  237  -  0.1  T.  (2GO) 

Let  the  initial  temperature  of  the  liquid  be  58°  F. ;  the 


272  HEAT    ENGINES.  [138.] 

initial  absolute  pressure  in  the  cylinder  60  Ibs.,  at  which  the 
temperature  will  be  239°  F.,  Eq.  (259)  ;  ratio  of  expansion, 
2  ;  back  pressure,  16  pounds  ;  and,  in  order  to  make  as  few 
other  assumptions  as  possible,  consider-the  law  of  expansion 
to  be 

p  <y¥    =  constant, 

and  since  the  ratio  of  expansion  is  small,  this  cannot  lead  to 
a  large  error. 
Then, 

p,  =  60  X  144  =  8640  Ibs.  per  sq.  ft. 


£4  (172),  ya=  2.18  X  2  —  4  °6  cu.  ft.  (262) 

Eq.  (177),  17=  2.18  x  8640  (10  -  8.35)  -  2304  X  4.3l>  =  21033         (203) 

ft.-lbs.  per  pound  of  naphtha  per  stroke,  or  26.40  thermal 
units. 

The  cut-oif  being  one  half,  the  weight  of  vapor  in  the 
three  cylinders  when  half  full  will  be 


X  37'5 


=  .01493  Ibs. 


2  X  1728  X  2.18 
Hence  per  pound  per  revolution  the  work  done  will  be 
21033  X  0.01493  =  314  foot-pounds. 

The  mean  work  per  revolution  in  the  preceding  experi- 
ment was 


which  results  agree  sufficiently  well  when  we  consider  that 
in  the  ideal  diagram  there  is  no  compression  or  clearance. 
Mean  effective  pressure,  theory,  (176), 

p9  =  60  (5  —  4.16)  —  16  =  34.4  Ibs.  per  sq.  in. 


[13y.]-  REMARKS.  273 

Mean  effective  pressure  from  the  experiment — 

330  (work  per  revolution) 
-^e       3jL?_ij_5  (piston  dixjjlacement) 

=  35.2  Ibs.  per  sq.  inch, 

which  is  a  very  fair  agreement  under  the  circumstances. 

The  heat  supplied  per  pound  will  be,  in  thermal  units, 
equations  (93)  and  (260), 

7i  =  c(Tl-  Tt)  +  A. 
=  0.434  X  181  -f  237  -  24  =  291.  (264) 

Efficiency  of  fluid,  (263),  (264)— 

2-~=  o^, 

which  we  will  call  nine  per  cent. 
Efficiency  of  furnace — 

°-033=0.36, 


or  36  per  cent ;  that  is,  36  per  cent  of  the  theoretical  heat  of 
combustion  of  the  naphtha,  as  determined  by  its  chemical 
composition,  is  utilized  by  the  boiler.  This  is  a  good  result 
for  so  small  a  boiler. 

139.  Remarks.  In  the  design  of  this  engine  about 
four  square  feet  of  heating  surface  per  horse-power  was 
allowed.  This  is  about  one  fourth  of  what  would  be  al- 
lowed in  the  design  of  a  steam  boiler.  In  the  former 
engine  the  boiler  is  completely  filled  with  the  flame  of  the 
burning  fluid,  and  thus  is  made  quite  efficient. 

The  naphtha  engine  has  met  with  much  favor  for  propel- 


274  HEAT   ENGINES.  [140.] 

ling  steam  launches  and  yachts.  Their  great  compactness  is 
apparent  from  the  preceding  report  ;  pressure  is  raised  very 
quickly  not  only  on  account  of  the  low  boiling  point  of 
naphtha,  but  especially  on  account  of  its  very  volatile  and 
highly  inflammable  character  when  used  for  fuel.  As  soon 
as  the  supply  is  cut  off,  the  flame  ceases,  and  no  vapor  is  left 
in  the  boiler  to  be  blown  off  or  cooled  down,  as  in  the  steam- 
engine.  As  a  fuel  it  is  more  conveniently  stored  in  pipes 
and  vessels  in  the  lower  part  of  the  boat  than  coal  can  be. 
It  appears  to  be  quite  as  efficient  as  a  steam  plant  of  this 
power— if  not  more  so.  But  every  power  has  disadvan- 
tages ;  and  this  is  objectionable  for  general  purposes  on  ac- 
count of  the  very  volatile  and  inflammable  character  of 
the  fluid,  endangering  as  it  might  all  combustible  material  in 
its  vicinity,  and  becoming  more  dangerous  the  greater  the 
quantity  stored  ;  still  it  is  especially  adapted  to  launches. 

14O.  Ammonia  engines  are  those  in  which  am- 
monia vapor  is  used  instead  of  steam.  These  engines  are 
condensing — a  condition  which  is  rendered  necessary  on  ac- 
count of  the  nature  of  the  substance.  Aqua-ammonia  is  in- 
troduced into  the  boiler ;  vapor  is  generated  in  precisely  the 
same  manner  as  steam,  and,  after  it  is  used  in  the  engine,  it 
is  condensed  and  pumped  back  into  the  boiler,  thus  using  it 
over  and  over. 

Much  interest  has  recently  been  excited  in  these  engines 
from  the  fact  that  ammonia  has  been  substituted  for  steam, 
using  the  same  boiler  and  engine,  and  doing,  it  is  claimed, 
the  same  work  with  less  fuel.  It  is  asserted  by  some  writers 
that  the  science  of  Thermodynamics  teaches  that  the  effi- 
ciency of  an  engine  is  independent  of  the  nature  of  the  work- 
ing fluid  when  used  between  the  same  limits  of  temperature ; 
and,  hence,  the  above  fact  leads  such  to  look  with  suspicion 
upon  the  correctness  of  this  part  of  the  science.  The  fact 
is,  the  bove  statement  is  correct  in  only  one  very  restricted 
case,  and  that  case  is  never  realized  in  practice  ;  we  will, 


[140.J  AMMONIA   ENGINES.  275 

therefore,  state  distinctly  some  of  the  principles  established 
by  this  science,  bearing  upon  this  part  of  the  subject. 

The  efficiency  of  an  engine  is  independent  of  the  fluid 
used  when  worked  between  the  same  limits  of  absolute  tem- 
perature, PROVIDED  ALL  THE  HEAT  RECEIVED  IS  AT  ONE  TEM- 
PERATURE AND  ALL  THAT  IS  REJECTED  IS  AT  ONE  LOWER  TEM- 
PERATURE, the  mass  of  fluid  in  the  engine  being  constant. 
(See  p.  161.) 

If  the  substance  could  be  worked  in  this  way,  steam,  am- 
monia, alcohol,  &c.,  would  be  equally  efficient.  The  fact 
that  there  is  a  latent  heat  of  evaporation  of  the  substance 
would  not  affect  the  truth  of  the  statement.  If  water  could 
be  worked  according  to  this  law,  beginning  with  a  tempera- 
ture of  60°  F.,  evaporated  at  250°  F.,  and  raised  to  300°  F., 
the  efficiency  would  be  the  same  as  if  the  substance  were  a 
perfect  gas,  or 

300  —60         AQ1, 
E=    300  +  460    ^ 

This  will  be  proved  for  a  vapor  in  the  Addenda,  Article 
112,  from  which  it  may  readily  be  inferred  to  be  true  for 
the  more  general  proposition. 

This  gives  the  absolute  maximum  efficiency  of  any  heat 
engine. 

But  no  engine  works  according  to  this  law.  In  practical 
vapor  engines,  the  mass  of  working  fluid  is  variable.  In 
such  engines,  it  has  been  shown  on  page  193,  Eq.  (o),  that  the 
effective  work  and  the  efficiency  both  depend  upon  the 
latent  heat  of  evaporation  and  the  specific  heat  of  the  sub- 
stance ;  hence — 

In  practice,  the  efficiency  of  all  vapor  engines  depends 
upon  the  nature  of  the  working  fluid,  and  involves  both  the 
latent  heat  of  evaporation  and  the  specific  heat. 

In  a  more  complete  theory  of  the  vapor  engine,  involving 
incomplete  expansion,  the  mean  specific  heat  of  the  fluid 
and  the  latent  heats  of  evaporation  at  the  lower  and  higher 


276  HEAT    ENGINES.  [140.] 

temperatures  are  involved  in  such,  a  complex  manner  as  to 
require  a  knowledge  of  their  numerical  values  iu  order  to 
determine  whether,  theoretically,  one  fluid  will  yield  a 
higher  efficiency  than  another. 

Again,  this  science  does  not  teach  that  the  efficiency  is 
independent  of  the  working  fluid  when  worked  between  the 
same  limits  of  pressure. 

To  illustrate  this  principle,  observe  that  in  Articles  110 
and  111,  the  work,  equations  (167)  and  (177),  is  expressed 
in  terms  of  pressure  only  ;  hence,  if  the  same  pressures  can 
be  obtained  at  lower  temperatures,  there  may  be  a  gain  of 
efficiency. 

Saturated  steam  at  100  Ibs.  per  sq.  in.  has  a  temp,  of  327°  F. 

«  a         ..       ],;      ..      a      ..       .-       ..    ..       .-         a     216°  F 

Dif.  TTT° 

Saturated  ammonia  vapor  at  100  Ibs.  per  sq.  in.  has  a 
temp,  of 57°  F. 

Saturated  ammonia  vapor  at  16  Ibs.  per  sq.  in.  has  a 

temp,  of —  23°  F. 

Dif.  ~SO° 

The  heat  expended  per  pound  of  vapor  above  the  lower 
temperature  will  be 
tor  steam,  Eq.  (184), 

H  =  111  «/+//.; 
and  for  ammonia 

H  =  80  <7+//e. 

If  the  indicated  work,  U,  is  the  same  in  both  cases,  since 
IIe  is  less  for  ammonia  than  for  water  and  C  also  less  than 
J,  the  efficiency  of  fluid  will  be  greater  for  the  former  than 
the  latter. 

Again,  the  formula  for  maximum  efficiency,  —     — ">  is 


[140.]  AMMONIA   ENGINES.  277 

applicable  only  to  a  constant  mass  of  fluid  when  working  in 
the  engine  /  and  is  not,  in  any  sense,  applicable  to  the  plant. 

The  efficiency  of  the  boiler  and  steam  connections  de- 
pends upon  the  absorption  of  heat  by  the  fluid,  radiation 
from  the  furnace,  and  radiation  from  the  connections.  It  will 
be  seen  from  the  preceding  remarks  that  a  given  pressure 
may  be  produced  with  ammonia  at  a  much  lower  temperature 
than  with  steam.  Such  being  the  fact,  a  lower  temperature 
may  be  maintained  in  the  furnace,  boiler,  and  connections, 
and  hence  less  heat  would  be  lost  by  radiation.  Our  knowl- 
edge of  ammonia  does  not  enable  one  to  determine  whether 
it  would  absorb  more  heat  in  the  same  time  than  would 
water;  if  it  would,  the  per  cent  of  heat  escaping  up  the 
chimney  would  be  less,  and  the  direct  efficiency  of  the 
boiler  thus  increased.  Admitting  that  an  ammonia  plant  is 
more  efficient  than  a  steam  plant,  we  see  that  this  science 
explains  the  cause.  But, 

Again,  this  science  teaches  that  a  condensing  engine  is 
more  efficient  than  a  non-condensing  one,  other  conditions 
being  the  same. 

In  the  cases  above  cited  that  have  come  to  the  knowledge 
of  the  author,  where  the  ammonia  plant  proved  superior  to 
the  steam  plant,  the  steam-engines  were  non-condensing, 
while  the  ammonia  engines  which  replaced  them  were  con- 
densing. Had  the  original  steam-engines  been  changed  to 
condensing  engines  there  would  have  been  an  increase  of 
efficiency ;  but  a  want  of  knowledge  of  certain  physical 
constants  of  ammonia  prevent  this  science  from  determin- 
ing with  certainty  whether  the  efficiency  would  have  been 
still  further  increased  by  substituting  it  for  steam. 

If  the  interest  on  the  cost  of  a  plant  and  the  cost  for  re- 
pairs be  considered,  a  small  plant  involving  a  condens- 
ing engine  may  not  be  as  economical  as  with  a  non- 
condensing  one,  although  the  former  may  be  the  more 
efficient. 


278  HEAT   ENGINES.  [141,  142.] 

Many  conditions  are  involved  in  the  choice  of  the  work- 
)*Jig  fluid,  aside  from  its  thermodynamic  relations :  as  the 
first  cost,  its  effect  on  the  working  parts,  its  safety  or 
danger  to  life,  and  the  character  of  the  necessary  mechanism. 

141.  A  biliary  vapor-eiigiiie  is  worked  by  means 
of  two  fluids,  one  more  volatile  than  the  other,  the  fluids 
being  worked  in  separate  cylinders.     If  a  surface  condenser 
of  a  steam-engine  be  cooled   by  ether,  the  ether    may  be 
vaporized  by  the  heat  given  up  by  the  steam,  and  made  to 
work  a  vapor-engine  in  precisely  the  same  manner  as  steam 
drives  a  steam-engine.     The  exhausted  vapor  of  the  vapor- 
engine  may  be  passed  through  a  surface  condenser  cooled  by 
water,  and  pumped  in  its   cooled  condition  into  the  con- 
denser of  the  steam-engine,  when  the  former  operation  may 
be  repeated. 

A  binary-engine  is,  theoretically,  no  more  efficient  than  a 
properly  designed  steam-engine  consuming  the  same  amount 
of  heat,  and  practically  is  not  as  economical  on  account 
of  unavoidable  waste  and  extra  cost  of  the  mechanism.  But 
a  steam-engine  wasteful  of  heat  may  be  improved  by  the 
addition  of  an  ether-engine.  (Manuel  du  Conducteur  des 
Machines  a  Vapeurs  conibinees,  M.  du  Tumbley,  Lyons, 
1850-51 ;  Institution  of  Civil  Engineers,  February,  1859.) 

142.  The    products   of  combustion    sometimes 
form,  the  working  fluid.     In  these  engines  the  entire  prod- 
ucts of  combustion  enter  the  cylinder.     The  principle    of 
their  analys'is  is  similar  to   the   Ericsson's   hot-air  engine. 
One  of  the  most  serious  objections  to  this  class  of  engines 
is  the  fact  that  the  solid  parts  of  the  working  fluid,  dust 
and  grit,  wear  out  the  working  parts,  with  which  they  come 
in  contact,  rapidly.     A  recent  attempt  has  been  made  by 
MM.  Bermier  Brothers,  Paris,  to  overcome  this  objection, 
but  as  yet  their  engine  is  only  an  experiment.     (Scientific 
American  Supplement,  1889,  p.  11099.) 


[143.1 


THE   INJECTOR. 


279 


THE  STEAM-INJECTOR. 

143.  The  injector  is  a  device  for  feeding  steam 
boilers,  in  which  steam  is  taken  from  the  boiler,  and,  by 
passing  through  the  instrument,  takes  water  with  it,  carrying 
the  water  and  condensed  steam  in  a  steady  stream  back 
into  the  boiler.  Fig.  69  shows  an  improved  form  of  one  of 
this  class  of  instruments.  A  valve  W  is  secured  to  the  rod 
£,  and  has  its  seat  on  another  valve  X.  A  is  a  tube  con- 
taining these  valves,  and  the  passage  of  steaui  through  the 


FIG.  69. 

tube  is  controlled  by  the  valve  X.  A  hollow  spindle,  be- 
ginning with  W  and  terminating  at  C,  passes  through  the 
valve  X,  and  may  be  moved,  independently  of  the  latter, 
for  a  short  distance,  by  means  of  the  lever  II,  thus  admitting 
steam  to  the  spindle  without  moving  the  valve  JT,  but  a 
further  movement  of  the  lever  will  unseat  the  latter  valve. 
The  chamber  M  M  contains  a  piston  JV  N,  which  terminates 


2SO  HEAT   ENGINES.  1144.J 

in  a  gradually  contracting  nozzle  at  a  point  0,  just  beyond 
C.  By  a  slight  movement  of  the  handle  II,  steam  issues 
from  the  orifice  O,  and  a  partial  vacuum  will  be  formed  in 
JV^Y,  into  which  water  will  be  forced  by  outside  pressure, 
and  then  forced  through  the  delivery  tube  D,  and  at  first  es- 
cape through  the  waste  orifice  jP,  and  as  soon  as  a  solid  stream 
escapes,  a  further  movement  of  the  lever  II  closes  the  orifice 
P  by  the  valve  A",  and  opens  the  valve  X,  and  a  continuous 
flow  of  water  will  then  pass  the  check-valve  into  the  boiler. 

If  too  much  water  passes  C  some  will  enter  the  chamber 
O  and  force  the  piston  N  N  back,  thus  throttling  the  water, 
and  if  sufficient  water  is  not  admitted  the  reduced  pressure 
at  0  will  cause  the  valve  to  move  forward  and  permit  more 
water  to  flow  in. 

144.  Theory  of  the  steam -inject  or. 

Let  7F0  =  the  weight  of  water  required  of  the  injector 
per  unit  of  time, 

TF  =  weight  of  steam  required  to  force  TF0  into  the 
boiler.  The  heat  in  the  steam  above  that  of  the  feed-water 
when  forced  into  the  boiler  will  be,  in  ordinary  heat  units, 
considering  the  specific  heat  of  the  water  as  uniform  and 
equal  to  unity,  equation  (134), 

Wh  =  [(rl-Tt)+xhe]  TF,  (265) 

and  this  will  be  the  heat  lost  by  this  amount  of  steam  in  the 
injector  and  which  is  assumed  to  be  imparted  to  the  feed- 
water. 

The  heat  imparted  to  the  water,  above  that  in  the  reser- 
voir from  which  it  is  taken,  will  be 

(r.  ~  rt)  TF0,  (266) 

where 

rt  =  the  absolute  temperature  of  the  feed-water  in  the  tank, 
ra  =  the  absolute  temperature  of  the  water  just  after  it  has 

passed  the  injector, 
T,  =  the  absolute  temperature  of  the  steam  in  the  boiler. 


[144.]  THEORY   OF   THE   STEAM-INJECTOR.  281 

It  will  be  shown  hereafter  that  the  work  of  lifting  the 
water  from  the  reservoir  to  the  injector  and  of  forcing  it 
into  the  boiler  together  require  only  a  small  fractional  part 
of  the  heat  energy  lost  by  the  steam  in  having  its  tempera- 
ture lowered  from  that  of  the  boiler  to  that  of  the  mixture 
of  steam  and  water ;  and,  neglecting  these  two  elements, 
expressions  (265)  and  (266)  become  equal,  giving,  in  terms 
of  ordinary  scales  of  temperature, 

TF  '~=  -T=T~+ah   W°'  (267) 

For  our  present  purpose  it  will  be  sufficiently  accurate  to 
assume  that  the  steam  supplied  to  the  injector  is  pure  satu- 
rated steam,  or  x  =  1,  and  that  equation  (77)  is  sufficiently 
exact,  or 

he  =  1114.4  -  0.7"  TV 

To  find  the  velocity  of  the  water  in.  the  passage  G,  Fig. 

70,  let 

p  =  the  absolute  pressure  per  unit  in  the  boiler, 

p0  =   ''          "  "  "      "     of  the  atmosphere, 

V=  the  velocity  of  the  water, 

tf  =  weight  of  unity  of  volume  of  the  water  =  62.4  per 
cubic  foot  at  ordinary  temperatures, 

then 

(268) 

The  value  of  6  may  be  found  with  sufficient  accuracy  by 
means  of  the  formula  at  the  foot  of  page  102,  thus 

d  =  —  -  - - ,  (269) 

v,       (_L_i_5o°y 

which,   for   150°  F.,    or   610°   absolute,    gives   d  =  61.2 
pounds,  and  this  value  might  properly  be  used  in  equation 


282  HEAT   ENGINES.  [144.] 

(268),  but  as  62.4  pounds,  the  weight  at  ordinary  tempera- 
tures, will  not  produce  an  error  of  1  per  cent  in  the  veloc- 
ity, and  as  by  its  use  the  resulting  formula  will  be  more 
generally  applicable  to  ordinary  cases,  we  retain  the  latter. 
Just  after  entering  the  chamber  G,  the  water  will  be  under 
atmospheric  pressure,  and  pa  =  2116.2  pounds  per  square 
foot,  and  2  g  =  64.4.  "With  these  values,  equation  (268) 
reduces  to 

F  =  1.0158  V~p~—  2116.2ft.  per  sec.  (270) 

If  p  be  in  pounds  per  square  inch, 


V  =  12.1896  V  p  -  14.7  ft.  per  sec.       .       (271) 
If  p  be  in  atmospheres, 


F  =  46.7355  tf  p  —  I  ft.  per  sec.  (272) 

If  the  diameter  of  the  suction  pipe  J^be  n  times  that  of 
the  passage  E,  the  velocity  in  it  will  be 

F,  =  £.  (273) 

To  find  the  area  of  the  opening  E 
for  the  passage  of  the  water ;  con- 
sider that  the  steam  passing  through 
the  injector  will  have  been  con- 
densed to  liquid  water,  then  will  the 
volume  of  the  water  and  condensed 
FIG.  70.  steam  passing  the  opening  per  sec- 

ond be  0.016  ( IF  +  TF0)  cubic  feet, 
and  if  k  be  the  area  of  this  section,  then 

j.  =   0.016  (  W  +  TT0) 

The  diameter  will  be 

(275) 


[144.  J  THEORY    OF   THE   STEAM-INJECTOR.  283 

To  fold  tJie  velocity  of  the  steam  issuing  from  the  eiid  of 
the  passage  C,  it  will  be  necessary  to  find  the  pressure  in 
the  condensing  chamber.  Let 

JP,  be  the  pressure  in  the  condensing  chamber  D, 

pv  the   pressure  of  the   atmosphere   on   the  water  in  the 

tank  B, 
h  =  C  B,  the  height  of  the  condenser  above  the  water  in 

the  tank, 
T7,,  the  velocity  of  the  water  at  F,  entering  the  condens- 

ing chamber, 
then 

17  =  2  g  [^=^  -  A],  (276) 

in  which  h  is  negative  because  the  water  is  raised  instead  of 
being  a  positive  head.     From  this  may  be  found 


The  velocity  of  the  steam  Fj  will  be  given  by  the  general 
equation  for  T7,  following  equation  (62),  page  82,  and  after 
substituting  for  T2  and  TI  their  values  in  terms  of  p  and  -y, 
becomes 


in  which  y  has  the  value  in  equation  (1-48),  page  152.     If 
the  steam  contains  no  moisture,  this  becomes 


/        r  fr>  \  0-H89~i 

F,  =  23.2687  V p  v  [I  -  ^2J  (279) 

The  area  of  the  cross-section  at  C  will  be 
T-F  Volume  of  steam  per  sec. 

~~T^~ 

and  the  diameter  will  be 


284  HEAT   ENGINES.  1 144.] 

The  work  done  l>y  the  injector  will  be  that  of  forcing  the 
mixture  of  steam  and  water  against  the  boiler  pressure  p 
sufficiently  far  to  make  a  displacement  for  "IF-J-  TT^,  pounds 
of  water.  Since  the  steam  will  be  subjected,  externally,  to- 
the  atmosphere  the  resultant  pressure  against  which  the 
water  is  forced  will  be  the  gauge  pressure,  or  p  —  j>0. 
Hence,  if  p  be  in  pounds  per  square  inch  the  work  will  be 

U  =  144  (p  -  p0}  (  TF  -f  TF0)  0.016  ft.  Ibs.      (282) 
The  efficiency  as  a  force-pump  will  be 

Work  done  U 


Heat  expended        (r,  —  ra  -\-  x  Ae)  TF  J ' 

The  efficiency  of  the  plant.  If  1  pound  of  coal  is 
equivalent  to  q  thermal  units,  and  w  pounds  are  required  to 
generate  TF  pounds  of  steam  from  the  temperature  of  the 
feed-water,  then 

F'  U 

E    =  —r , 


and  if  all  the  heat  of  the  coal  could  be  utilized  for  generat- 
ing steam  and  the  steam  were  pure  saturated,  E'  would  be 
the  same  as  E.  But  there  is  always  a  waste  of  heat  in  the 
furnace  and  boiler.  If  the  q  thermal  units  would  evaporate 
n  pounds  of  water  at  and  from  212°  F.,  if  there  were  no 
waste  of  heat,  and  in  an  actual  boiler  a  pound  of  coal  did 
evaporate  n^  pounds  under  the  same  conditions  ;  then  if  T< 
be  the  temperature  of  the  feed-water  and  H  the  total  heat 
of  steam  at  the  temperature  Tl  of  the  boiler,  then 

TF(H  —  T4)  =  -$66  n,  w  =  966  n  •  ^  w  =  qw  -'  ;  (285) 


The  value  of  n  may  be,  theoretically,  from  11  to  15,  de- 
pending upon  the  composition  of  the  coal,  and  nl  from  6  to 


[144.]  THEORY   OF   THE   STEAM-INJECTOR.  285 

11,  depending  upon  the  composition  of  the  coal  and  the  effi- 
ciency of  the  furnace. 

The  duty  will  be  the  work  done  per  100  pounds  of  coal, 
or, 


D  =  100  X  144  X  gauge  pressure  X 


Volume 
of  water 

injected 
per  Ib.  of 

steam. 


Pounds 
of  (steam 
evapo- 
rated per 
Ib.  of  coal. 


Effect  of  rejecting  the  work  of  raising  the  water  and  of 
forcing  it  into  the  boiler  in  the  above  analysis. 

In  the  following  exercise  it  will  be  seen  that  if  the  gauge 
pressure  be  90  pounds  per  square  inch,  and  other  conditions 
as  there  given,  there  will  be  expended 

(331  —  120  -f  794)  X  0.05  =  50 

thermal  units  in  supplying  0.833  pounds  of  water  to  the 
boiler.  To  raise  this  weight  of  water  20  feet  by  suction — 
a  distance  too  large  to  be  realized  in  practice — would  re- 
quire 

0.833  X  20  -=--  778  =  0.02 

thermal  units,  which  is  only  ^Vir  °f  ^^?  an(^  hence  may  be 
omitted  in  the  computation. 

The  work  of  forcing  0.833  pounds  into  the  boiler  will  be, 
equation  (282),  in  thermal  units, 

U  =  144  X  90  X  0.833  X  0.016  -=-  778  =  0.22, 

which  is  also  so  small  compared  with  50  that  it  may  be 

omitted.     The   theory  above   given,  in  which   these   two 

items  are  omitted,  is,  then,  sufficiently  accurate  for  engi- 
neering purposes. 

EXERCISE. 

If  the  steam  pressure  in  a  boiler  is  90  pounds  gauge  per 
square  inch,  height  of  suction  4  feet,  and  the  boiler  is  re- 
quired to  make  3000  pounds  of  steam  per  hour  ;  required 


286  HEAT    ENGINES.  U**-] 

the  area  of  the  section  k  of  the  passage  E  'for  the  water,  the 
velocity  of  the  steam  F",  at  6',  the  diameter  of  the  suction- 
pipe,  its  section,  being  5  times  that  of  the  section  k  (which 
is  an  average  of  actual  values),  the  steam  containing  10  per 
cent  of  moisture,  the  feed-water  in  the  tank  being  60°,  the 
temperature  of  the  mixture  of  water  and  condensed  steam 
120°  before  it  is  forced  into  the  boiler  ;  also  the  ratio  of  the 
velocity  of  the  steam  to  that  of  the  water,  and  the  weight 
of  water  to  that  of  the  steam. 

We  have 

p  =  104.7,  h  =  4,  r«  -  60°,  Ta  =  120°,  n  =  5. 

From  steam,  table,  or  equation  (81),  page  97,  find  T,  =  330°.9  F. 

TT.  =  ^.  =  0.833  Ibs.  per  second. 
36WJ 

h,  =  1114.4  -  0.7  X  330.9  =  882.6,  Eq.  (77). 
an  =  0.9  ;  .  '  .  J-,  A   =  794.34. 


V  =  12.1896  V90  =  115.63  ft.,vel  of  water  at  E  and  G,  Fig.  70,  Eq.  (271). 
k  _  0.016  (0.05  +  0.833)  _  0>00123  sq  ft  _  0.017568sq.  in.,  Eq.  (274). 

115.63 
d  =  0.149  in.,  diameter  of  water  passage  EOT  0,  Eq.  (275). 


p,  =  2116.2  -  4  X  62.4  -  4-4"  1845'87  lbs"  P61"  **•  ft' 

=  12.818  Ibs.  per  sq.  in.  at  F,  Eq.  (277). 
v  =  4.217,  volume  of  one  pound  of  steam  at  104.7  Ibs.,  Eq.  (86). 

Vl    =  23.2687  1/104.7  X  144  X  4.217  |~1  -  /.18818  \o-"""| 

=  2572.5  ft,  per  sec.  velocity  of  the  steam  at  C.  Eq.  (278). 

Ft  =  4.22  X  0.05  X  144  i-  2572.5  =  0.0110  sq.  in.,  Eq.  (280). 

rf,  =  0.12,  diameter  of  steam  nozzle,  Eq.  (281). 
Vtl.  offiteam  _  2572.3  _ 
Vel.  of  water  ~  115.63  ~~ 

No  allowance  has  been  made  in  this  computation  for  contraction  or. 
f  fictional  resistances,  and  hence  the  diameters  must  be  made  larger  than 
here  found  in  order  to  deliver  the  assumed  amounts.  The  diameters 
should  be  about  1.1  to  1.2  times  those  here  found. 


[144.] 


THEOKY    OF   THE   STEAM-1XJECTOK. 


237 


U=  144  X  90  X  (0.833  -f  0.05)  X  0.016  =  183.099  ft.  Ibs.,  Eq.  (282). 

=  0.235  thermal  units. 


E  = 


0.235 


- — -  =  0.0046,  Eq.  (283),  or  the  efficiency  is  less  than  one- 


1005  X  0.05 
half  of  one  per  cent. 

If  10  pounds  of  coal  evaporates  1  pound  of  water  at  and  from  212° 
F.,  it  will  require,  under  the  conditions  of  this  exercise,  to  evaporate 
0.05  pound  of  water, 


w  =     •  -  =  0.00581  11,,  Eq   (285). 

966  X  10 

If  the  coal  be  equivalent  to  pure  carbon,  it  would  evaporate,  with- 
out loss  of  heat,  14500  +  966  =  15  Ibs.  at  and  from  212°  F.,  and  if  one 
pound  in  the  plant  actual!}'  would  evaporate  10  Ibs.,  then  would  the  effi- 
ciency of  the  plant  be 


E'  = 


183-°" •  1?  =  0.00418  X  -  =  0.00279,  Eq.  (286)- 

778   X  0.05  (1183  -  60)   13  3 


Duty  =  1296000  X  (16.6  X  0.016)  x  10  =  3442176  ft.  Ibs.,  Eq.  (287). 


TABLE  I. 

GIVING  CERTAIN  RELATIONS  WHEN  THE  DELIVERY  INTO  THE  BOILER  IS 
1  POUND  OP  WATER  PER  SECOND,  NEGLECTING  LIFT  AND  WORK  OF 
FORCING  THE  WATER  INTO  THE  BOILER  ;  TEMPERATURE  OF  THE 
FEED-WATER  BEING  60°  F.,  AND  OF  THE  MIXTURE  AND  STEAM  BE- 
FORE ENTERING  THE  BOILER  160°  F. 


Gauge 
Pressure. 

Diameter 
of  Steam 
Nozzle  in 
Inches. 
Eq.  (281.) 

Diameter 
of  Water 
Nozzle  in 
Inches. 
Eq.  (275.) 

Velocity 
of  Steam, 
Ft.  per 
Second. 
Eq.  (279.) 

Velocity 
of  Steam 
andWater, 
Ft.  per 
Second. 
Eq.  (271.) 

Ratio  of 
Velocities. 
Col. 
(4)  -*•  (5) 

Ratio  of 
Weight  of 
Water  to 
Steam. 
Eq.  (267.) 

Ratio  of 
Volume  of 
Steam  to 
Water. 

30 

0.28 

0.21 

2007  9 

66.7 

30. 

10.3 

55.9 

40 

0  24 

0.20 

2178.8 

77.1 

28. 

10.3 

46.2 

50 

0.22 

0.19 

2213.5 

86.2 

25. 

10.4 

39.4 

60 

0  20 

0.18 

2428.8 

94.4 

25. 

10.5 

34.4 

70 

0.18 

0.178 

2522.3 

101.2 

25. 

10.5 

30.4 

80 

0.17 

0.172 

2554.1 

108.0 

24. 

10.5 

37.6 

90 

0.167 

0.166 

2590.6 

115.6 

22. 

10.5 

25.2 

100 

0.159 

0.160 

2735.8 

121.8 

22. 

10.5 

22.8 

120 

0.142 

0.154 

2842.7 

133.5 

21. 

10.6 

19.6 

140 

0.133 

0.149 

2922.3 

144.2 

20. 

10.6 

17.2 

160 

0  127 

0.143 

2999.7 

154.2 

19. 

10.6 

15.3 

(1) 

(2) 

(3) 

(4) 

(5) 

(6) 

(7)       1      (8) 

288 


HEAT   ENGINES. 


As  the  temperature  160°  F.  of  the  mixture  of  steam  and  water  is  near 
the  higher  limit  of  reliable  working  of  the  injector,  we  take  another  case 
of  lower  temperature. 

TABLE    II. 

GIVING  RESULTS  FROM  THE  SAME  DATA  AS  FOR  TABLE  I  ,  EXCEPT 
THAT  THE  TEMPERATURE  OF  THE  MIXTURE  IS  ASSUMED  TO  BE 
140°  F. 


Gauge 
Pressure. 

Diameter 
of  Steam 
Nozzle  in 
Inches. 
Eq  (281  , 

Diameter 
of  Water 
Nozzle  in 
Inches. 
Eq  (275) 

Velocity 
of  Steam, 
Ft.  per 
Second. 
Eq.  (279.) 

Velocity 
of  Steam 
andWater, 
Ft.  per 
Second. 
Eq.  (271.) 

Ratio  of 
Velocities. 
Col. 
(4)  -«•  (5). 

Ratio  of 
Weight  of 
Water  to 
Steam. 
Eq.  (267.) 

Ratio  of 
Volume  of 
Steam  to 
Water. 

30 

.25 

.18 

2030 

66.7 

30.4 

13.21 

44.2 

40 

.218 

.202 

2211 

77.1 

28.0 

13.29 

36.3 

50 

.197 

.190 

2326 

86.2 

27.0 

13.29 

31.1 

60 

.158 

.180 

2431 

94.4 

25.6 

13  33 

27.1 

70 

.150 

.175 

2526 

101.2 

24.9 

14.52 

22.0 

90 

.146 

.164 

2677 

115.6 

23.1 

13.42 

19.6 

100 

.137 

.160 

2743 

121.9 

22.5 

13.60 

17.7 

120 

.125 

.152 

2845 

133  5 

21.3 

13.49 

15.4 

140 

.116 

.147 

2934 

144  2 

20.3 

13.53 

13.5 

160 

.108 

.14^ 

3008 

154.2 

19.5 

13.58 

12.0 

(1) 

(2)            (3)     i      (4) 

(5) 

(6) 

(7) 

(8) 

145.  Approximate  Formulas.  Certain  general 
inferences  will  be  apparent  if  we  assume  average  condi- 
tions. 

Let  T4  =  60°  F.,  Tt  =  150°  R,  and  x  =  1,  then,  equation 

TF0  _  964.4    ,      1 


|_  _±_  T 

W          90    ^300     '* 


(288) 


If  the  gauge  pressure  be  80  pounds,  then  T7,  =  323°  F., 
and 


W 


=  11.79; 


that  is,  under  ordinary  conditions,  1  pound  of  steam  will 
inject  about  12  pounds  of  water  into  the  boiler;  or  13 
pounds,  including  its  own  weight. 


[146.]  INJECTOR   COMPARED.  289 

"For  the  diameter  of    the  Cylindrical   water-passage,    E, 
equation  (275), 


/0.016  X| 

12.1896       /;  (289) 


that  is,  the  diameter  will  vary  directly  as  the  square  root  of 
the  weight  of  water  injected  per  second,  and  inversely  as 
the  fourth  root  of  the  gauge  boiler  pressure. 

The  velocity  of  the  steam,  according  to  the  preceding 
tables,  will  be  about  half  a  mile  per  second. 

The  velocity  of  the  water  will  be  about  100  feet  per  sec- 
ond. 

The  duty  will  be,  if  gauge  pressure  =  80,  and  9  pounds 
of  steam  be  generated  per  pound  of  coal, 

D  =  1152000  X  12  X  0.016  X  9  nearly  =  2000000  nearly. 

Since  there  will  be  some  frictional  resistance  and  radiation, 
and  since  9  pounds  of  water  are  rarely  evaporated  at  80 
pounds  gauge,  the  duty  would  be  somewhat  less  than 
2000000. 

Efficiency,  equation  (283), 


which  is  about  -^  of  1  per  cent.     The  efficiency  of  the 
plant  would  be  about  -£-  of  1  per  cent  as  a  pump. 

146.  Injector  compared  with  Direct-Acting 
Pump.  By  comparing  these  results  with  those  on  page 
182  it  will  be  seen  that  the  efficiency  and  duty  of  the  in- 
jector are  much  less  than  that  of  a  direct-acting  pump  —  being 
about  ^  as  efficient.  This  is  for  service  as  a  pump.  But  as 
a  heat  device,  if  there  be  no  radiation  nor  lift  of  feed-water 
the  efficiency  of  the  injector  will  be  perfect  ;  similarly,  if 


290  HEAT   ENGINES.  [146.] 

all  the  exhaust  heat  from  the  direct-acting  pump  be  re- 
turned to  the  boiler,  and  there  be  no  radiation,  the  heat 
efficiency  of  the  pump  will  also  be  perfect ;  and  hence  in 
either  case  would  cost  nothlntj  for  fuel.  In  both  cases  the 
furnace  (or  boiler)  heats  the  water  from  the  temperature  of 
the  feed  to  that  of  the  boiler.  If  there  be  no  losses  from 
radiation,  the  difference  in  the  cost  for  fuel  in  running  the 
two  devices  will  be  that  which  furnishes  the  steam  for  run- 
ning the  pump  for  doing  the  same  work,  if  this  steam  be 
wasted  at  the  exhaust.  To  illustrate :  the  work  done  by  1 
pound  of  steam  in  the  approximate  cases  above  is  that  of 
forcing  13  pounds  of  water  against  30  pounds  pressure, 
and  is 

U  —  144  X  80  X  13  X  0.01  o  =  2396  ft.  Ibs. 

One  pound  of  steam  in  the  direct-acting  pump  will,  at 
about  70  or  80  Ibs.  boiler  pressure,  do  the  actual  work  of 

10,000  foot-pounds ; 
hence,  to  do  2396  foot-pounds  will  require 

2396  -T-  10000  =  0.24  Ibs.,  nearly, 

of  steam  ;  hence,  it  requires,  in  this  case,  about  24  hun- 
dredths  as  much  steam  to  feed  the  boiler  with  a  direct-act- 
ing pump  as  with  an  injector.  But  this  steam  is  saved  by 
the  injector,  and,  we  assume,  is  wasted  by  the  pump.  If 
1  pound  of  coal  generate  S£  pounds  of  steam  under  a 
pressure  of  80  Ibs.  gauge,  this  waste  will  require  0.24  -=-  8.5 
=  0.0282  pounds  of  coal  for  every  12  pounds  of  feed-water 
forced  into  the  boiler.  To  evaporate  this  12  pounds  of 
water  will  require  12  -f-  8.5  =  1.41  pounds  of  coal ;  hence, 
the  fractional  part  of  the  fuel  required  by  the  pump  will  be 

0.0282  -f-  1.41  =  0.02, 
or  about  2  per  cent  of  the  fuel  burned  in  the  furnace. 


[146.]  INJECTOR   COMPARED.  291 

Or,  if  the  engine  requires  30  pounds  of  feed-water  per 
horse-power  per  hour,  it  will  require 

0.0282  X  n  =  0.0705 

pounds  of  coal  to  work  the  feed-pump  per  hour  per  horse- 
power ;  and  if  the  plant  requires  3£  pounds  of  coal  per  horse- 
power per  hour,  then  will  the  fractional  part  of  the  fuel  re- 
quired by  the  feed-pump  be 

=  0.0201, 


35000 

or  about  2  per  cent  of  the  fuel  burned,  as  before  found. 

The  low  efficiency  of  the  injector,  as  a  pump,  is  due  to 
the  fact  that  the  high  velocity  of  the  steam  is  very  suddenly 
reduced  to  a  comparatively  low  one  by  its  impact  against 
the  non-elastic  water,  and  the  kinetic  energy  lost  by  the 
steam  will  be  as  the  difference  of  the  squares  of  the  velocity 
before  impact  and  that  after. 

Considering  the  velocity  of  the  steam  as  25  times  that  of 
the  mixture,  and  the  weight  of  the  mixture  as  13  times  that 
of  the  steam,  the  kinetic  energy  of  the  mixture  will  be 

13  G>V)2  =  0.0208 

of  the  initial  energy  of  the  steam  ;  or  98  per  cent  of  the 
initial  energy  is  lost  by  the  change  of  velocity  at  E.  The 
2  per  cent  remaining  is  gradually  diminished  on  account 
of  the  decreasing  velocity  in  the  passage  from  G  to  V. 

The  thermodynamic  theory  of  the  inspirator  is  the  same 
as  that  of  the  injector. 

Steam-injectors  are  also  used  as  pumps  where  intermit- 
tent action  is  required,  as  in  the  hold  of  a  ship,  and  in 
mines  ;  also  as  ejector-  condensers  when  attached  to  the 
escape-pipe  of  a  condensing  engine  to  avoid  the  use  of  an 
air-pump  ;  also  as  a  gas-pump  where  it  was  more  efficient 
than  as  a  water-pump  ;  also  as  a  steam-blower  ;  and  also  in 
the  well-known  case  of  the  locomotive  exhaust. 


THE   PULSOMETER. 


[147.] 


THE   PULSOMETER. 

147.  The  Pulsometer  is  a  pump  consisting  prin- 
cipally of  two  bottle-shaped  chambers,  A,  A,  joined  together 
side  by  side,  with  tapering  necks  bent  toward  each  other, 
uniting  in  one  common  upright  passage,  into  which  a  small 
ball,  C,  is  fitted  so  as  to  oscillate 
with  a  slight  rolling  motion  be- 
tween seats  formed  in  the  junc- 
tion. 

These  chambers  also  connect  by 
means  of  openings  with  the  verti- 
cal induction  passage,  D,  having 
valves,  E,  E,  and  their  seats,  F,  F. 
The  delivery  passage,  II.  which 
is  common  to  both  chamber*,  is 
also  constructed  so  that  in  the 
openings  that  communicate  with 
each  cylinder  are  placed  valve- 
seats  fitted  for  the  reception  of  the 
same  style  of  valves,  G,  G,  as  in  the  induction  passage. 

J  represents  the  air  chamber,  cast  with  and  between 
the  necks  of  chambers  A,  A,  and  connects  only  with  the  in- 
duction passage  below  the  valves  E,  E. 

A  small  brass  air  check-valve  is  screwed  into  the  neck  of 
each  chamber,  A,  A,  and  one  into  the  vacuum  chamber  «/, 
so  that  their  stems  hang  downward.  Those  in  the  chamber 
allow  a  small  quantity  of  air  to  enter  above  the  water,  to 
prevent  the  steam  from  agitating  it  on  its  first  entrance. 

Conceive  that  the  left  chamber  is  full  of  water;  steam 
passes  to  the  left  of  the  valve  (7,  and  acting  by  its  pressure 
directly  upon  the  upper  surface  of  the  water,  forces  the 
water  through  the  valve  G  and  into  the  air  chamber  J. 
During  this  operation  the  chamber  A  is  being  filled,  and 
water  by  its  momentum  finally  drives  the  valve  C  to  the 


FIG.  71. 


[148.]  HEAT  ENGINES.  293 

left,  thus  cutting  off  steam  communication  with  the  kfc 
chamber  in  which  the  steam  condenses,  forming  a  vacuum, 
when  water  will  be  forced  through  the  valve  E  by  at- 
mospheric pressure  into  the  left  chamber,  while  the  steam 
is  forcing  the  water  out  of  the  right  chamber.  All  the 
steam  entering  the  pump  is  condensed  and  forced  out  with 
the  other  water;  and  the  temperature  of  the  discharged 
water  will  be  higher  than  that  entering  the  pump. 

148.  Analysis.  The  work  done  by  the  pulsometer 
will  be  that  of  lifting  the  water  from  the .  source  to  the 
pump  by  the  operation  usually  called  u  suction,"  and  of 
lifting  this  water  and  the  condensed  steam  to  the  point  of 
delivery — neglecting  losses,  such  as  friction,  contractions, 
etc. 

Let 

TT0  be  the  weight  of  water  raised  in  a  unit  of  time, 
TTr,  the  weight  of  steam  used  in  the  same  time, 
T^  the  temperature  of  the  water  at  the  source, 
Tlt    "  "  "     "    mixture, 

r,     "  "  "     "    steam, 

Ae,  the  latent  heat  of  evaporation  of  the  steam, 
A0,  the  height  of  the  pump  above  the  source, 
A,,    "        "       "     "   delivery  above  the  pump, 
A,  the  total  height  =  7?0  +  A,. 

Considering  the  specific  heat  of  water  as  constant  and 
equal  to  unity,  the  heat  lost  by  the  steam  will  oe 

W  (T-Tt  +  A.), 
and  the  heat  gained  by  the  water  will  be 

^8  (2\  -  r.), 

and  no  allowance  being  made  for  radiation,  these  will  be 
equal ; 


294  THE   PULSOMETER.  [148.] 


(301) 


Observing  the  boiler  pressure,  and  the  temperature  of  the 
water  before  and  after  mixture,  the  ratio  of  the  weight  of 
the  steam  to  that  of  the  water  may  be  determined. 

The  work  will  be 

U=   TT.A.  +  (TT.  +  IT)/*,  (302) 

If  the  temperature  of  the  feed-water  be  the  same  as  that 
of  the  source,  or  T0,  then  will  the  heat  expended  be 

H  =  J  W(T  -  Tt  +  Ae)  ;  (303) 

hence  the  efficiency  will  be 

E-  U  -  W'**  +  (W*  +  F)A'  (304 

-H-     JW^-T.+h.)' 

If  the  work  of  lifting  the  condensed  steam  and  f  notional 
resistances  be  neglected,  then 

'(305) 


EXERCISES 

1.  By  actual  measurement  105000  gallons  of  water  were 
raised  in  ten  hours  with  274  pounds  of  coal  a  height  of  38 
feet,  and  drawn  horizontally  600  feet.  If  10  per  cent  be 
allowed  for  resistances,  find  the  work  done  in  ten  hours,  the 
weight  of  water  raised  per  pound  of  coal  and  the  horse- 
power ;  and  if  a  pound  of  coal  evaporated  7^-  pounds  of 
water,  find  the  pounds  of  coal  required  per  horse-power  per 
hour,  the  weight  of  water  raised  per  pound  of  steam,  the 
increase  of  temperature  of  the  water  pumped,  assuming  its 
initial  value  to  be  60°  F.,  the  gauge  pressure  50  pounds,  and 
the  efficiency,  the  feed-water  also  being  60°  F. 

If  a  gallon  be  231  cubic  inches,  and  a  cubic  foot  be  62.2 
pounds,  then 


[148.]  HEAT  ENGINES.  295 

Weight  of  water,  105000  X  T2TVe  X  62.2  Ibs.  =  873000. 

Work  for  10  hours,  873000  X  38  X  1.10  ft.  Ibs.  =  36491400. 

"       "     1  hour,        .         .....         .         .  3649140. 

Horse-power,          .         .        .    '•  „         .         .  1.84. 

Coal  per  horse-power  per  hour,  Ibs.,    .         .         .  14.8. 

Water  raised  per  pound  of  coal,  Ibs.,      .       . ...  3186. 

Pounds  of  steam,  274  X  7$,       .         .         .         .  2000. 

Water  raised  per  pound  of  steam,  Ibs.,    .         .  436.5. 

Work  done  per  pound  of  steam,  ft.  Ibs.,      .      .  -  *  18246. 

Heat  in  the  steam  above  60°  F.,  B.  T.  IL,      ,  1137. 

Increased  temp,  of  water,  1137  -j-  436.5,  Deg.  F.,  2.6. 

Efficiency, 1?587__  =  .  0.0187 

J'  1137  X  778 
Efficiency,  Eq.  (305),        .   -.      ....         .        0.0180 

or  less  than  2  per  cent.    (See  page  452.) 

The  assumption  in  regard  to  the  evaporating  power  of 
the  furnace  would  make  the  efficiency  of  the  furnace 
about  56  per  cent,  making  the  efficiency  of  the  entire  plant 
over  1  per  cent. 

Diameter  of  discharge  pipe,  if  the  coefficient  of  discharge 
be  0.8  and  velocity  4  feet  per  second, 


d  = 


A  three-inch  pipe  was  used. 

Pressure  producing  a  velocity  of  5  feet  per  minute 
against  the  atmosphere  and  a  head  of  38  feet  of  water, 
pounds  per  square  inch " 31 

2.  If  the  temperature  of  the  source  be  60°  F.,  of  the 
mixture  65°  F.,  the  gauge  pressure  60  pounds,  lift  by  suction 
5  feet  and  lift  above  the  pump  15  feet;  required  the.number 
of  pounds  of  water  raised  per  pound  of  steam,  the  efficiency  ; 
also  the  horse-power  if  300  pounds  of  steam  are  used  per 
hour.  (These  quantities  are  ideal.) 


COMPRESSED    AIR-ENGINE. 


[149,  150.] 


COMPRESSED  AIR-ENGINE. 

149.  A  Compressed  Air-Engine  is  an  engine  in 
which  the  working  fluid  is  common  air  under  a  high  tension. 
The  air  is  usually  compressed  by  a  machine  called  an  air- 
compressor,  to  a  tension  of  from  40  to  1000  pounds  to  the 
square  inch,  and  stored  in  an  air  reservoir,  called  a  receiver, 
from  which  it  is  taken  for  driving  an  engine.     Any  ordinary 
steam-engine  may  be  run  by  compressed  air  ;  the  only  prac- 
tical difficulty  being  the  tendency  of  the  moisture  in  the  air 
to  freeze,  and  thus  choke  the  exhaust.     The   freezing  may 
be  prevented  by  causing  a  circulation  of  warm  air  about  the 
exhaust  passage  through  channels  especially  provided  ;  and 
without  this  the  evil  may  be  mitigated  in  a  measure  by  a 
proper  form  of  the  exhaust  passage  —  gradually  enlarging  it 
as  it  goes  outward  —  and  making  it  smooth,  so  that  the  ice,  if 
formed,  will  not  adhere  so  firmly. 

150.  Analysis.     We   assume    that   the    cylinder   is 
filled  with  air  of  uniform  pressure  and  temperature  up  to 
the  point  of  cut-off,  that  then  it  expands  according  to  an 
assumed  law,  then  exhausts  and  a  uniform  back  pressure 
during  the  back  stroke  ;  also  that  there  is  no  clearance.    The 

diagram  cleared  from  irregularities  and 
clearance  will  be  similar  to  A  B  C  E 
FA,  Fig.  72. 

Let  jp,  be  the  absolute  pressure  O  A, 
Pv  the  absolute  pressure  //  C  at  the 

end  of  expansion, 
p»  the  absolute  back  pressure  HE, 
v,  the  volume  of  a  pound  of  air  at  the  pressure  pa 


B 

K    J^fc 

Q 

FIG.    72. 

(a)  Adiabatic  expansion  —  incomplete. 
per  pound  at  full  pressure  will  be 


The  work  done 


v,. 


[150-1  HEAT   ENGINES.  297 

The  work  done  per  pound  during  expansion  will  be,  Ex- 
ercise 3,  page  64,  or  the  second  equation  in  Article  56, 

T*  d  r  --=   Cv  (TI  -  r,). 
The  negative  work  during  the  return  stroke  will  be 


hence,  in  the  cycle,  the  work  done  per  pound  will  be 
U  =  A  B  GEF  =  Cv  (Tl  -  T.)  +Pl  Vl  -  p,  Vv      (306) 

Since  the  fluid  is  considered  perfect,  we  have,  equations 
(2)  and  (29), 

p,  v,  =  R  r,  =  (Cv  -  Cv}  *„ 
p.  v,  =  R  r2  =  (#,  -   <7V)  r2  ; 

.  •  .  V  =  Cv  (r,  -  r2)  +  (C\  -  Q  (r,  -  r^.         (307) 

(J)  Adiabatic  expansion — complete.     The  back  pressure 
will  be  along  CD,  and  j93  =  p»  r^  =  r3,  and 

17  =  A  B  CD  =  Op  (r1  —  rs)  =  Jy  cv  (T,  -  rs).     (308) 
(c)  If  there  be  no  expansion,  p^  =  p»  Tt  =  ra,  and 

U=(Cp-  Ov)  ('l  -  ^)  r,.  (309) 

Equations  (307)  and  (309)  may  be  put  in  a  more  symmetrical  form  by 
introducing  an  auxiliary  TX,  thus  : 

C^  (r,  -  r,)  +  (<7P  -  Q  (^  -  rag)  =  C,  (r,  -  r,)  ; 

...    I:  _£'+3L^£.£sJ  +  LZl-1£.  (310) 

ft^V  ^     -      ^        ^  7      ^8 

Equation  (309)  will  reduce  to  precisely  the  same  value  ;  hence  (307) 
and  (309)  become 

U=  <7Pl  -         T,.  (311) 


298 


COMPRESSED    AIR-ENGINE. 


[150.] 


Making  a  table  of  values  of  — ,  having  for  argument  ^?,  the  computation 

TI  pt 

for  the  work  may  be  much  facilitated.     Let  y  —  1.41,  then  : 


g 
p» 

Ta 

Pt 
Pi 

Ta 

2 

0.855 

9 

0  742 

3 

0.807 

10 

0.739 

4 

0.783 

11 

0.736 

5 

0.768 

12 

0.734 

6 

0.758 

13 

0.732 

7 

0.732 

14 

0.731 

8 

0.746 

15 

0.730 

Final  temperatures.     Knowing  the  initial  temper- 
ature, the  final  may  be  found  from  equation  (41),  which  is 


-MS)1 


(312) 


FINAL  TEMPERATURES,  THE  INITIALS  BEINO  r,  =  68°  P.  OB  r,  =  528V66. 


PI 
Pt 

Final  Temp.  Deg.  F. 

Pi 
Pt 

Final  Temp.  Deg.  F. 

2 

-  28 

9 

-181 

3 

-  76 

10 

-189 

4 

-107 

11 

-197 

5 

-129 

12 

-203 

6 

-148 

13 

-209 

7 

-160 

14 

-214 

8 

—  171 

15 

-219 

Such  low  temperatures  are  fatal  to  successful  working  if 
moisture  be  present  in  the  working  air,  as  ice  would  be 
formed  in  the  exhaust.  Either  the  initial  temperature  must 
be  considerably  higher  or  the  range  of  pressures  must  be 
small,  or  adiabatic  expansion  avoided,  unless  the  air  be 
thoroughly  dry. 

(d)  Let   the  expansion   l>e   isothermal    and    incomplete. 


[150.]  HEAT   ENGINES.  299 

Then 


(313) 
and 

U,  =  A  B  C  E  F  =  pl  vl  -f-   i  2)  d  v  —  ps  v^ 

=  Pi  Vi  [~1  +.  I0g^  _  Pi].  (314) 
L  v,      p,_\ 

For  the  same  cut-off,  the  isothermal  will  lie  wholly  above 
the  adiabatic,  and  L\  in  equation  (314)  will  exceed  U  in 
equation  (306).  In  order  to  secure  isothermal  expansion  it 
is  necessary  to  heat  the  air  in  the  cylinder  during  expan- 
sion. It  is  not,  however,  practical  to  maintain  a  uniform 
temperature  in  this  way.  The  very  low  final  temperature  has, 
however,  in  practice  been  prevented  by  working  the  air  in 
one  cylinder  through  a  part  of  the  full  range  of  pressures, 
then  exhausting  into  a  receiver  and  there  heating  it,  after 
which  it  is  worked  in  a  second  cylinder. 

(e)  'Expansion  isothermal  and  complete.  In  this  case 
the  terminal  pressure  H  (7,  Fig.  72,  will  equal  the  back 
pressure  H  E\  or  p9  =  p^  in  equation  (314)  ; 

.-.  U  =  DAB  C  =  G  B  OH 

=  p,  v,  log*  r  =  122.5  rt  loglo  r.     (315) 

Weight  of  air  per  minute.  Let  W  be  the  number  of 
pounds  of  working  air  necessary  to  deliver  N  horse-powers 
per  minute,  then,  sinee  V  is  the  work  in  foot-pounds  per 
pound  of  air, 

W  U  =  33000  N-, 


Volume  of  the  cylinder.  If  pM  -y0,  be  respectively  the 
pressure  and  specific  volume  of  the  air  before  compression, 
and  if  the  temperature  of  the  air  entering  the  engine  be  the 
same  as  before  compression,  then  p^  v,  =  p0  va>  and  the  final 


300  COMPRESSED   AIR-ENGINE.  [150.] 

specific  volume  v,  may  be  found  when  the  law  and  amount 
of  expansion  are  fixed.  The  terminal  volume  of  TF  pounds 
will  be  TF  #„  and  this  will  equal  the  volume  swept  through 
by  the  piston  per  minute,  if  there  be  no  clearance.  Let  V 
be  the  volume  of  the  cylinder  and  n  the  number  of  single 
strokes  of  the  piston  per  minute,  then  for  a  double-acting 
engine, 

In  V=   Wv,  =    WE  A; 


.      F=  88000  a 

2  n  Up, 

Efficiency.  In  order  to  determine  the  efficiency,  the  full 
cycle  of  operations  must  be  known,  and  this  involves  the 
law  of  compression,  which  will  be  considered  in  the  discus- 
sion of  the  air-compressor.  We  know,  however,  if  air  were 
compressed  according  to  any  law  and  expanded  according 
to  the  same  law,  there  being  no  escape  of  heat  by  radiation 
between  the  states  of  expansion  and  compression,  that  the 
efficiency  would  be  unity  ;  but  there  would  be  no  resultant 
work,  even  neglecting  the  friction  of  the  engine. 

The  above  formulas  being  for  perfect  conditions  must  be 
modified  in  order  to  conform  to  practice.  Pernolet  deter- 
mined that  the  moisture  in  the  air,  when  converted  into  va- 
por, did  not  materially  affect  the  theoretical  results  of  con- 
sidering the  air  as  dry.  The  weight  of  air  as  determined 
from  equation  (316)  must  be  increased  to  allow  for  clear- 
ance, leakage,  and  imperfect  working,  as  is  done  with  the 
steam-engine  ;  and  this  must  be  still  further  increased  in 
determining  the  weight  of  air  before  it  enters  the  compres- 
sor, to  allow  for  the  imperfect  working  of  the  compressor. 

Compressed  air-engines  are  frequently  used  where  if 
steam  were  used  there  would  be  excessive  condensation,  as, 
in  mines  and  other  underground  work,  for  driving  drills, 
pumps,  hoisting  engines  and  locomotives  ;  also  for  small  in- 
termittent powers. 


[151.]  HEAT    ENGINES.  "801 

THE  COMPRESSOR. 

151.  All  air-compressor  is  a  kind  of  air-pump  for 
receiving  air  at  ordinary  conditions,  and  after  compressing 
it  to  a  higher  tension,  forcing  it  into  a  vessel  called  a  re- 
ceiver. It  is  not  a  motor,  but  requires  a  motor  for  driving 
it.  The  principles  of  construction  are  substantially  the 
same  as  for  an  hydraulic  pump,  although  in  detail  clear- 
ance spaces  must  be  as  small  as  possible  and  the  valves  be 
so  made  as  to  work  with  certainty.  The  valves  are  the  most 
important  details,  and  have  received  a  large  amount  of  at- 
tention from  inventors  and  practical  men.  The  best  condi- 
tion for  the  proper  working  of  the  air  valves,  both  inlet  and 
exit,  is  to  have  them  open  and  close  by  moving  vertically 
and  automatically  ;  and  for  this  reason  the  compression  cyl- 
inder has  often  been  placed  vertically,  although  vertically 
moving  valves  are  used  with  horizontal  cylinders.  In  the 
latter  case,  at  least  -two  valves  at  each  end  of  the  cylinder 
are  commonly  used — one  for  inlet,  the  other  for  outlet. 
When  the  cylinders  are  vertical,  the  compression  cylinders 
are  frequently  single  acting,  and  are  driven  by  a  double- 
acting  steam  cylinder.  The  steam  cylinder  may  be  vertical 
or  horizontal.  In  some  cases  the  axes  of  the  cylinders  have 
been  inclined  to  each  other,  but  the  horizontal  types  are 
most  common.  Other  fluids  than  air  may  be  compressed 
in  such  a  compressor. 

Fig.  T3  is  a  view  of 
a  duplex  air-compressor 
made  by  the  author  and 
worked  in  a  silver  mine 
in  Colorado.  The  two 
steam  cylinders  are  at 
the  left  hand,  and  the 
other  two  are  the  com-  FIG- 

pression  cylinders. 

The  cranks  are  so  set  that  the  steam  in  one  cylinder  will 


302 


THE   COMPRESSOR. 


[152.] 


be  at  full  pressure  when  the  piston  in  the  air  cylinder  on 
the  other  side  will  be  near  the  end  of  its  stroke  where  re- 
sistance is  greatest. 

152.  Analysis.  During  the  back  stroke  of  the  piston 
the  air  flows  into  the  cylinder  ;  assume 
that  it  has  the  uniform  pressure  0  D, 
Fig.  74.  During  the  return  stroke  the 
pressure  rises  from  C  to  B,  and  the  air 
is  then  forced  into  a  receiver  at  a  press- 
ure which  we  assume  to  be  uniform 
and  equal  to  O  A. 


FIG.    74. 


Let  p9't  vt'  rj  represent  state 
P\->  v\i  Ti         "  " 


the   subscripts  denoting 


the  states  ordinarily  used  in  this  work,  and  the  accents  dis- 
tinguishing them  from  the  notation  of  an  engine. 

a.  Adiabatic  compression.     The  work  will  be,  equation 
(308), 

Ur  =  ABCD=C»  (r/  -  r,').  (318) 

For  air  Cv  =  184.77  (p.   53).     We  have,  equation  (42), 
page  61, 

(319) 


where  y  =  1.4.     From  this  the  final  temperature  due  to 
compression  may  be  found.     Thus  : 

FINAL  TEMPERATURES,  THE  INITIAL  TEMPERATURE  =  68°  F.,  OR  r 


ML 

Pt1 

Final  Temp. 
Deg.F/ 

£ 

Final  Temp. 
Deg.  F. 

2 

186 

0 

538 

3 

266 

10 

569 

4 

329 

11 

599 

5 

382 

18 

625 

6 

427 

13 

650 

7 

468 

14 

675 

8 

505 

15 

700 

[152.]  HEAT   ENGINES.  303 

The  temperature  exceeds  that  of  boiling  water  under  the 
corresponding  pressure  before  the  tension  reaches  four 
atmospheres. 

b.  Isothermal  compression.     Here/>/  •v/  =  pj  v,'  ;  and 

CV  =  A  B  C  D  =  pi'  «/  -  p,'  «,'  +  /"*,'*  pdv  =  122.5  r,  Iog10  ^<  (320) 

which  is  equation  (315).     Equations  (31  8),  (319),  (320),  give 

— 


which  is  less  than  unity  for  all  practical  cases  ;  hence  iso- 
thermal compression  requires  less  work  than  adidbatic  com- 
pression between  the  same  pressures. 

Isothermal  compression  is  secured  approximately  by  in- 
jecting water  into  the  cylinder  in  the  form  of  a  fine  spray 
in  sufficient  quantity  to  absorb  the  heat  due  to  compression. 
The  same  result  is  secured  less  efficiently  by  performing 
part  of  the  compression  in  one  cylinder,  and  allowing  the  air 
to  cool  in  a  receiver,  after  which  the  compression  is  com- 
pleted in  another  cylinder.  If  air  is  compressed  adiabati- 
cally,  the  heat  lost  between  the  compressor  and  the  motor 
represents  lost  energy.  If  no  heat  were  thus  lost,  adiabatic 
compression  would  be  desirable. 

Energy  lost  by  radiation.  If  TI  be  the  final  temperature 
in  the  compressor  and  T3  the  temperature  at  the  motor,  then 
will  the  energy  lost  per  pound  be 

<?P  (*,  ~  rt).  (322) 

Weight  of  water  to  he  injected  in  order  to  reduce  the 
temperature  a  given  amount.  In  order  to  secure  isothermal 
compression,  the  refrigerant  would  necessarily  be  indefi- 
nitely large,  for  otherwise  its  temperature  would  be  raised. 
Let  W  be  the  pounds  of  water  which  must  be  injected  to 


304  THE    COMPRESSOR.  [152.] 

reduce  the  temperature  of  one  pound  of  air  from  T*  to  T°, 
the  temperature  of  the  water  being  raised  from  Tt°  to  7\v. 
Since  the  specific  heat  of  water  will  be  considered  as  unity, 
we  have 

-  Tt)=  CtW-Tj.  (323) 


Volume  of  the  compressing  cylinder.  If  there  were  no 
losses,  the  volume  of  the  compressor  cylinder  would  be  the 
same  as  that  of  a  compressed-air  motor  doing  the  same  work 
as  expended  in  compression,  working  under  the  same  law  of 
expansion.  Hence,  if  V  be  the  volume  of  the  compression 
cylinder,  W  the  pounds  of  air  compressed  by  N'  horse- 
powers per  minute,  U'  the  work  necessary  to  compress  one 
pound  under  an  assumed  law  and  force  it  into  a  receiver,  n' 
the  number  of  single  strokes  of  the  engine,  and  r/the  initial 
temperature  of  the  air,  then  equation  (317)  gives 

F  =  33000  N'  R  r,'  __  W  R  rt' 
2n'  Up*  '    2  nr  p%'  ' 

Modifications.  The  initial  pressure  in  the  cylinder,  p^ 
will  be  less  than  that  of  the  external  air,  for  the  valves  will 
offer  some  resistance  to  the  inflow  of  air,  and  it  Would  take 
a  short  time  to  establish  equilibrium,  and  the  temperature  of 
the  cylinder  may  expand  the  air.  If  there  be  any  clear- 
ance, all  of  the  compressed  air  would  not  be  forced  into  the 
receiver.  For  these  reasons,  and  also  on  account  of  the  heat 
lost  by  radiation,  the  volume  of  the  cylinder  should  be  con- 
siderably larger  than  that  found  from  equation  (324).  This 
would  be  secured  by  assuming  the  horse-powers,  JV7,  ex- 
pended in  compressing  the  air  as  proportionately  larger  than 
the  horse-powers  N,  to  be  delivered  by  the  motor,  but  so 
many  contingencies  arise  in  practice  that  a  definite  rule  can- 
not be  stated  beforehand.  Deficiency  in  size  in  the  con- 
struction may  often  be  overcome  in  practice  by  increasing 
the  piston  speed. 


[153-]  HEAT   ENGINES.  305 

153.    Efficiency    of   compressor  and    engine.     For 
complete  adiabatic  expansion,  equations  (308)  and  (318)  give 


T  i 

=  —  ;,  approx. 


'1  _    "  ' 

L         r/J 


This  operation  may  be  illustrated  by  Fig.  75.  The  air 
will  be  taken  into  the  compressor  at 
the  absolute  temperature  TZ'  at  (7,  then 
compressed  adiabatically  to  temperature 
r'  at  B,  then  forced  into  the  receiver  at 
the  constant  temperature  r/and  pressure 
£>/•  The  work  done  by  the  compressor 


per  pound  of  air  will  be  /  C 'B II.     The  FIG.  75. 

air  then  loses  heat  and  enters  the  engine  at  A  under  the  press- 
ure pl  =j9j 'and  a  temperature  TJ  and  expands  adiabatically  to 
D,  the  temperature  being  reduced  to  r^  where  it  is  exhausted. 
The  work  of  the  engine  will  be  ID  A  H.  The  resultant 
work  will  be  A  B  C  D.  If  no  heat  were  lost,  the  tempera- 
ture at  A  would  equal  that  at  B,  and  that  at  D  equal 
that  at  C,  or  rl  =  r/y  r.2  =  rj  ;  .  • .  E  =  1,  or  the  efficiency 
would  be  perfect.  In  this  case,  however,  A  D  will  fall  on 
B  C,  and  the  resultant  work  will  be  zero.  The  compression 
might  be  along  D  A  and  expansion  along  B  C.  Equation 
(325)  expresses  the  efficiency  if  the  air  enters  the  motor  at 
a  less  pressure  than  that  of  HA,  and  exhausts  it  at  a  higher 
or  lower  pressure  than  that  of  C.  In  this  case  the  cycle 
will  not  be  complete. 

Mass  of  fluid  constant.  In  some  operations,  especially 
in  refrigerating  machines,  the  mass  of  working  fluid  is 
constant,  the  operation  B  A  being  effected  by  abstracting 
heat,  and  D  C  by  supplying  heat.  In  this  case,  if  A  D 
and  B  C  are  adiabatics,  the  heat  supplied  along  D  C  will 
be,  per 


306  THE  COMPRESSOR.  [134.1 

#,  =  c»  (*;  -  o, 

and  abstracted  along  B  A, 

//,=  C9(r;-Tfo 
also 

H*     '  ;  1 

and  the  efficiency  of  fluid  will  be 

-F_  CKr.'-rJ-CKT.'-r,)          r/  -  r/     (326, 
=  X~  #p  (*/-*,)  ~~^~" 

the  same  as  for  the  perfect  elementary  engine  ;  and  is  the 
fraction  of  the  work  which  is  transmuted  into  heat.  If  the 
operation  were  in  a  reversed  direction,  the  result  would  be 
positive  and  would  be  the  fraction  of  the  heat  absorbed 
which  would  be  transmuted  into  work. 

154.  Friction  of  air  in  pipes.  The  experiments 
at  Mont  Cenis  gave  the  formula 

7^=0.00936^,  (327) 

a 

in  which  d  is  the  diameter  of  the  pipe  in  inches,  I  the 
length  in  feet,  n  the  velocity  in  feet  per  second,  and  F  the 
loss  of  pressure  in  pounds. 

EXERCISES. 

1.  Required  the  volume  of  the  cylinder  of  a  double-act- 
ing air-compressor  making  50  revolutions  per  minute  to 
deliver  to  a  compressed  air-engine,  making  100  revolutions, 
sufficient  air  to  give  5  indicated  horse-powers,  allowing  fifty 
per  cent  lost  in  the  power  of  the  compressed  air.  Let  the 
initial  temperature  of  the  air  be  60°  F.,  and  compressed  iso- 
thermally  to  5  atmospheres  absolute,  the  initial  pressure  in 
the  engine  also  5  atmospheres  at  the  same  temperature. 


[154.]  HEAT  ENGINES.  307 

Find  the  weight  of  water  which  must  be  injected  per  stroke 
at  55°  F.,  that  the  temperature  may  not  exceed  65°  F.  ;  the 
volume  of  the  cylinder  of  the  engine,  the  point  of  cut-off 
of  the  engine  that  the  expansion  shall  be  complete,  the 
final  temperature  at  the  end  of  the  stroke,  and  the  efficiency 
of  the  system. 

First  find  the  work  which  one  pound  of  compressed  air 
will  do.     We  have 

TI  =  60  +  460  =  520,  Pl  =  14.7  X  5  =  73.5,  p,  =  14.7. 

Final  temperature,  r,,  520  X  (i)?.  degrees  absolute  .............  328. 

degreesF  .................................  —132. 

Difference  of  temperatures,  r,  —  r.,  ..........................  192. 

Work  per  Ib.  of  air,  Eq.  (308),  184.77  X  192,  ft.-lbs  ............  35476. 

Work  required  per  minute,  33000  X  5,  ft.-lbs  ..................  165000. 

Air  required  per  minute,  Ibs  ............  .....................  4.65. 

Vol.  cyl.,  Eq.  (317),  4  65  X  53.21  X  328  4-  (200  X  2116.3)cu.  ft-  0.192. 

Diameter,  if  stroke  is  li  times  the  diameter,  inches  ............  6.55. 

Ratio  of  expansion,  Ei  =  (^Yr4  =  (5)*  .....................       3.15. 

Vl          \PJ    ' 

Air  to  be  supplied  to  compressor,  4.65  X  2,  Ibs.  per  minute  ____       9.30. 
*  per  stroke,  Ibs  ...............      0.093. 

Vol.  air  cyl.,  Eq.  (324),  ™*™™  '  ^  *  ..............       ^ 


Diameter,  if  stroke  is  1^  times  the  diameter,  inches  ............  12.1. 

Work  per  Ib.,  Eq.  (320),  122.5  X  520  X  0.699,  ft.-lbs  ..........  44526. 

Work  per  minute,  44526  X  9.3,  ft.-lbs  ........................  414092. 

Efficiency,  165000  -f-  414092  =  ...............................  0.40. 

Water  injected,  0.24  X  192  -j-  10,  Ibs.  per  Ib.  of  air  .............  4.61. 

Ibs.  per  stroke  4.61  X  0.093  ...................  0.43. 

cu.  in,  per  stroke  0.43  X  0.016  X  1728  .........  11.89. 

If  the  temperature  be  limited  to  100°  F.,  it  would  require 
less  than  three  cubic  inches  of  water  per  stroke. 

2.  In  the  preceding  exercise  if  the  compression  were 
adiabatic,  find  the  final  temperature  of  compression,  the 
final  pressure  being  5  atmospheres,  initial  temperature  of 
the  air,  65°  F.  ;  also  the  efficiency. 


308  STEAM   TURBINES.  [156.] 


THE  STEAM  TURBINE. 

155.  Steam  turbines  act  on  the  same  general  prin- 
ciples as  hydraulic  turbines ;  an  essential  difference  bei:ig 
that  water  is  considered  non-compressible,  while  steam  ana 
other  vapors  are  compressible.     A  more  general  term  for 
this    class   of   turbines    would   be   elastic  vapor  turbines. 
They  may  be  reacting,  like  the  Barker  mill,  Whitelaw  or 
Scottish  turbine,  parallel  flow,  outward  or  inward  flow.    One 
is   described   in    the   Pneumatics  of  Hero  of  Alexandria. 
Rankine  also  mentions  an  inward-flow  turbine  which  was 
used  at  the  Glasgow  City  Saw  Mills,  and  was  considered 
equal   in    efficiency   to   an   ordinary  high-pressure    engine 
(Steam  Engine,  p.  538).     The  claim,  however,  is  not  sus- 
tained by  any  authentic  experiments.     Very  few  of  these 
turbines  appear  to  have  been  in  use  until  quite  recently  ;  now 
they  are  being  used  to  drive  electrical  dynamos,  chiefly  on 
account  of  the  very  small  space  occupied  by  them  and  the 
ease  with  which  they  may  be  located  wherever  desired.     In 
many  cases  they  are  wasteful  of  steam  on  account  of  the 
clearance   spaces  permitting  a  part  of   the  steam  to  pass 
through  the  engine  without  doing  work,  but  one  quite  re- 
cently invented  by  Messrs.  Dow  appears  to  be  a  great  im- 
provement on  previous  engines  of  this  class. 

156.  Balanced  outward-flow  steam  turbine. 

The  turbine  shown  in  Fig.  76  is  the  joint  invention  of  J.  II. 
Dow  and  H.  II.  Dow,  of  Cleveland,  Ohio.  A  A  represents 
the  casing,  or  stationary  part  of  the  engine  ;  B  B  the  rotat- 
ing wheel  firmly  secured  to  the  shaft  C,  and  containing  the 
buckets  or  floats  shown  in  the  section  D  D,  which  are  ar- 
ranged in  concentric  circles  ;  and  concentric  with  these  and 
between  them  are  rings  projecting  from  the  stationary  part 
of  the  engine  through  which  are  cut  steam  passages  or 
guides.  Steam  entering  through  the  stationary  part  at  E, 


[is?.; 


HEAT   ENGINES. 


309 


FIG.    76 


passes  both  sides  of 
the  rotating  disk 
FF,  into  the  an- 
nular cavity  near 
the  centre,  thence 
outward  through 
passages  in  the  an- 
nular spaces  be- 
tween th<3  buckets, 
and  through  the 
buckets,  finally  es- 
caping at  the  outer 
circumference  at  D, 
and  is  conducted 
away  at  the  exhaust 
at  G.  . 

It  is  balanced  laterally  by  means  of  the  disk  F  F,  which 
is  firmly  secured  to  the  shaft  C\  so  that  if  there  be  a  lateral 
movement,  however  small,  the  space  on  one  side  of  the  disk 
will  be  reduced  and  on  the  other  side  enlarged,  so  that  the 
increased  amount  of  steam  entering  the  latter  will  force  it 
back  to  its  normal  position.  It  is  claimed  that  this  move- 
ment may  be  limited  to  0.002  of  an  inch.  The  energy  of 
the  steam  is  gradually  absorbed  by  the  wheel  as  it  passes 
through  it,  thereby  diminishing,  its  pressure  and  .causing  ex- 
pansion, similar  to  that  in  a  multiple-expansion  engine, 
there  being  six  compoundings  in  this  wheel. 

157.  Analysis.  Reaction  Turbine.  These  may  be 
constructed  like  the  Barker  mill,  Scottish  or  Whitelaw  tur- 
bines, or  other  hydraulic  turbines  of  this  class.  The  section 
of  the  orifices  is  very  much  smaller  than  that  of  the  arms. 
The  reaction  of  the  steam  as  it  escapes  from  the  arms  im- 
parts to  them  a  rotary  motion,  and,  consequently,  as  the 
fluid  passes  outward  in  the  arms  a  rotary  motion  is  imparted 


310  STEAM  TURBINES.  [157.] 


to  it  in  common  with  the  arms  ; 
the  fluid  escaping  in  a  backward 
direction  relative  to  the  motion 
of  the  orifices.  The  velocity  of 
exit  will  depend  upon  the  three 
elements  : 

1.  The  pressure  at  the  orifices  due  to  the  boiler  pressure, 
as  if  the  arms  were  at  rest ; 

2.  The  additional  pressure  due  to  rotation,  as  if  the  orifices 
were  closed ;  and 

3.  The  velocity  of  the  orifice  relative  to  the  earth. 

The  velocity  of  discharge  relative  to  the  earth  will  be  the 
resultant  due  to  these  three  causes  acting  simultaneously. 
On  account  of  the  compressibility  of  the  fluid  and  the  cen- 
trifugal action,  the  density  of  the  steam  will  increase  from 
the  axis  of  rotation  outward.  The  centrifugal  force  of 
the  liquid,  if  any,  in  the  vapor  will  cause  the  liquid  to  flow 
outward  more  rapidly  than  the  vapor,  and  thus  greatly  com- 
plicate the  solution ;  and  it  would  be  still  further  compli- 
cated by  considering  the  change  of  temperature  and  of  re- 
evaporation  in  passing  outward  along  the  arm.  We  will 
assume  that  the  steam  is  dry  saturated  or  slightly  super- 
heated and  the  temperature  uniform. 

Let  A,  be  a  head  producing  the  pressure  at  the  entrance 
to  the  arms  where  the  weight  of  a  unit  of  volume  is  wl  and 
pressure  p.t.  At  any  distance  p  let  the  pressure  be  p  and 
w  the  weight  of  unity  of  volume ;  then,  since  the  weight 
will  be  directly  as  the  pressure,  we  have 

pl  =  A,  wa          p  =  A,  w.  (328) 

The  variation  of  pressure  will  be  due  to  the  centrifugal 
force  of  an  element  whose  thickness  is  d  p  and  base  unity ; 

(329) 


[157.]  HEAT   ENGINES.  311 

where  co  is  the  angular  velocity  per  second.     These  equa- 
tions give 


Fa* 

~P,  (330) 


where  p,  is  the  pressure  at  the  orifice,  V3  the  velocity  of  the 
orifice,  and  Fthe  velocity  due  to  the  head  hr  If  F,  =  0, 
pl  =  p^  as  it  should.  The  pressure  due  to  the  centrifugal 
force  will  be 


The  interior  pressure  ready  to  produce  velocity  will  be 
p,  ;  now,  if  the  orifice  be  opened  into  the  atmosphere,  the 
resultant  pressure  will  be  j?a  —  p^  when  pn  is  the  pressure  of 
one  atmosphere.  The  velocity  of  exit  will  be  found  from 
equation  (278),  page  283,  after  making  y  =  1.3,  as  given  in 
equation  (145),  page  151  ;  hence 


(332) 

Without  rotation,  the  velocity  relative  to  the  orifice,  or 
the  earth,  would  be 

/  T  /  />•>  \0.2307~| 

F,  =  16.705  A/  p>  v,  [  1  -  [£^       J  .  (333) 

The  orifices  have  a  velocity  ca  p  =  F,  opposite  to  Fa  ; 
hence  the  velocity  of  discharge  relative  to  the  earth  will  be 

F=F,  -F3.  (334) 

The  pounds  of  steam  discharged  per  second  will  be,  equa- 
tion (64),  value  of  R,  p.  103  ; 


312  STEAM   TURBINES.  [157.] 


W=  wtksV'=  1.8295  **JP,(^}    T  ^rA       (335) 

"in  which  T&  is  the  coefficient  of  discharge,  pl  the  pressure  in 
the  arm  at  theoriiice  and  isj9,  in  equation  (330),  ra  the  tem- 
perature outside  and  r,  the  temperature  in  the  arm,  wy  the 
weight  of  unity  of  volume  at  the  section  of  greatest  con- 
traction, and  W  the  weight  discharged  per  second  at  that 
point. 

The  work  done  by  the  reaction  per  second  will  be 


or  per  pound, 


U=l(V,-Vt)  F,. 


9 

The  energy  expended  will  be  that  in  the  steam  above  the 
temperature  of  the  feed-water,  and  per  pound  will  be, 
equations  (93)  and  (77), 

H  =  778  (1114.4  +  0.305  T,  —  TJ,  (337) 

where  T^  is  the  temperature  at  the  boiler  in  degrees  Fahr- 
enheit and  Ty  that  of  the  feed. 

The  efficiency  will  be 

The  horse-powers  will  be 

W  77 

H  P  =  (339) 

33000 

EXERCISE. 

1.  In  a  reaction  turbine  having  orifices  12  inches  from 
the  axis  of  rotation,  if  the  boiler  pressure  be  50  pounds 
(gauge),  section  of  all  the  orifices  0.02  of  a  square  inch,  coeffi- 


t157-]  HEAT   ENGINES.  313 

cient  of  discharge  .50,  velocity  of  the  orifices  one  fourth  the 
theoretical  velocity  due  to  the  steam  pressure  when  the 
engine  is  at  rest;  find  the  work  per  second,  the  efficiency, 
the  temperature  of  the  feed-water  being  60°  F.,  the  horse- 
power and  the  pounds  of  water  used  per  horse-power  per 
hour. 

From  Eq.  (84)  or  from  steam  tables,  volume  of 

1  Ib.  of  steam  at  64.7  Ibs.  absolute,  will  be,  cu.  ft.  6.58. 

Weight  of  a  cu.  ft.,  Ibs 0.152. 

Yel.  of  steam,  no  rotation  (333),   F,  ft.  per  sec. . .  2223. 

Telocity  of  orifices  £  of  2223,  Fs 555.7. 

Revolutions  per  second 89.3. 

"              ''     minute 5358. 

Pressure  at  axis  where  steam  enters  the  arms 64.7. 

Pressure  at  orifices,  (330),  p.»  Ibs.  per  sq.  in 140. 

Tol.  of  1  Ib.  steam  at  orifices  6.58  X  64.  T  ~  140,  3.04. 

Toi.  discharge  relative  to  orifice,  F2,  (332) 2639. 

"               "        "  the  earth,  V,  -  F3 2084. 

Work  per  pound,  U,  (336),  ft.  Ibs 35952. 

Energy  expended,  H,  (337),  (T,  =  297,  T,  =  60)  890732. 

Efficiency  (338),  per  cent 4.0. 

Discharged  per  sec.  if  T,  =  220°,  Eq.  (335),  Ibs.,  0.0195. 

Horse-powers 1.2,7. 

Steam  per  H.  P.  per  hour,  Ibs 55. 

This  result' appears  to  make  the  engine  quite  as  efficient 
as  small  non-condensing  engines.  Theoretically,  the  efficien- 
cy would  be  high  if  the  velocity  of  the  orifices  were  half 
that  of  the  theoretical  velocity  of  discharge,  or  twice  that 
assumed  above,  but  the  resistance  of  the  air  to  the  motion 
of  the  arms  would  be  so  great  as  to  consume  the  power  of 
the  engine.  At  the  velocity  in  this  example  the  speed  of 
the  orifices  is  over  seven  miles  per  minute.  At  200  feet  per 
second,  or  about  ten  times  the  average  highest  piston  speeds, 
the  efficiency  would  be  very  low,  not  even  considering 


314 


STEAM   TUKBINES. 


[158.] 


prejudicial  resistances.     To  be  efficient  the  speed  must  be 
high. 

158.  Outward-flow  turbine  of  Fig.  76.  The 
best  speed  for  the  turbine  requires  that  the  fluid  shall  be 
discharged  with  the  least  velocity — just  sufficient  to  escape 
from  the  wheel.  To  accomplish  this  the  steam  must  ex- 
pand down  nearly  to  the  pressure  of  the  atmosphere. 

If  the  wheel  is  so  constmcted  and  operated  that  the  steam 
will  expand  without  transmission  of  heat,  the  method  of 
Article  112  will  be  applicable,  and  work  done  per  pound 
of  steam  would  be 


(340) 


if  there  were  no  losses  from  friction,  contraction,  eddies  or 
clearances. 

To  determine  the  speed  requires  a  definite  knowledge  of 
its  construction.  A  properly  constructed  wheel  must  run 
at  a  definite  speed  for  maximum  efficiency,  and  it  cannot  be 
correctly  analyzed  for  speeds  differing  much  from  that,  on 
account  of  eddies  or  whirls  being 
induced,  the  effect  of  which  cannot 
be  formulated. 

Let  0,  Fig.  78,  be  the  centre  of 
the  wheel,  a  g  the  inner  rim,  5  e  the 
outer,  c  a  a  guide,  a  I  a  bucket.  If 
there  were  no  friction  or  eddies,  the 
analysis  for  several  concentric  circles 
of  buckets  would  be  the  same  as  if 
all  the  work  were  done  in  one  series 
of  buckets  ;  so  we  treat  the  case  as 
if  the  several  series  were  devel- 
oped into  one.  Let  the  initial  ele- 


[158.  j  HEAT   ENGINES.  315 

ment  of  the  bucket  at  a  be  tangent  to  the  radius  0  af*  of 
the  wheel,  a  d  a  tangent  to  the  inner  rim,  a  h  tangent  to  the 
guide  c  a  ;  also 

d  a  h  =  of,  the  angle  between   the   terminal  element  of  a 
guide  and  the  inner  rim  of  the  wheel, 
r,  =  O  a,  the  inner  radius  of  the  wheel, 
F",  the  velocity  of  the  steam  at  a  in  the  direction  a  ^, 
-yt,  the  tangential  component  of  F, 
VT,  the  radial  component  of  F", 

and  the  same  letters  with  accents  to  indicate  similar  quanti- 
ties at  £>,  the  point  of  discharge  ;  also  &>,  the  angular  velocity 
of  the  wheel,  and  M  =  TF  -r-  </,  the  mass  of  steam  flowing 
through  the  wheel  per  second  ;  then 

Y  cos  a  =  vt,         vt  tan  a  •=.  vr ; 
V  cos  a'  =  v/,       vt'  tan  a.'  =  v/. 
The  rim  velocities  will  be,  respectively, 
r  &?,         r'  oo  . 

According  to  the  principle  of  mechanics — the  difference 
of  the  moments  of  momenta  of  the  fluid  upon  entering  and 
quitting  the  wheel  into  the  angular  velocity  of  the  wheel 
equals  the  energy  imparted  to  the  wheel — we  have  for  the 
work  per  second, 

M  (vt  r  —  vt'  r')  o>, 
or  per  pound, 

Z7  =  fo  r -</•')  | ,  (341) 

and  this  should  be  approximately  the  value  of  equation 
(340).  In  the  former,  much  the  greater  part  of  the  heat  of 
the  steam  is  lost  at  the  exhaust,  while  in  the  latter  consider- 
able energy  may  be  lost  in  the  kinetic  energy  of  the  steam 

*  In  the  Dow  engine  the  buckets  are  straight  and  the  angle  at  a  acute. 


316  STEAM   TURBINES.  [158.] 

as  well  as  that  in  the  exhaust.  In  order  that  there  be  no 
impact  on  entering  the  wheel,  we  have  vt  =  w  r,  and  if  the 
steam  were  discharged  from  the  wheel  radially,  t\f  r'  would 
be  zero  ;  as  a  rough  approximation  let  it  be  zero,  then 


.•.  car  =    V  g  L  . 

Let  n  be  the  ratio  of  the  actual  to  the  theoretical  value  of 
U,  then  _ 

07  >•  =    V  n  g  U.  (34-3) 

On  account  of  condensation,  clearance  and  friction,  n  for 
non-condensing  engines  is  from  £  to  ^. 

If  the  wheel  does  not  run  at  best  velocity,  vtf  r'  will  not 
be  zero  ;  let  it  be  rj  t\  r,  in  which  ij  will  have  a  different  value 
for  every  different  velocity  ;  also  GO  r  will  not  equal  vif  let  it 
be  £  G>  r  ;  then 


J 


The  number  of  revolutions  per  minute  will  be 

JT=8i»         •  -  (846) 

EXERCISES. 

1.  Consider  a  Dow  steam  turbine  run  with  steam  at  70 
pounds  boiler  pressure  (gauge),  using  600  pounds  of  steam 
per  hour,  the  efficiency  being  one  fourth  the  theoretical. 

Assume  that  the  gauge  pressure  at  the  engine  is  67  pounds, 
or  about  4  per  cent  less  than  the  boiler  pressure,  that  the 
terminal  pressure  is  17  pounds  per  square  inch,  the  inner 
radius,  r  =  1J.  We  have 


[158.]  HEAT   ENGINES.  317 

r,  =  773,        TS  =  679,        Hei  =  778  X  S94£  =  696000, 
ut  =  21.9,  eq.  (150),  -M,  -f-  v,  =  4.05  =  ratio  of  expansion. 
Work  per  Ib.  of  steam,  Eq.  (340),  ft.  Ibs.,  approx  .  .     88000. 
Work  per  Ib.  as  per  hypothesis,  ft.  Ibs  ............     22000. 

Steam  per  H.  P.  per  hour,  1980000  -f-  25075  Ibs  .  90. 

Yel.  of  inner  rim  if  rj  =  0.  1,  n  =  £,  e  =  4,  ft.  per  sec.         223 
Revolutions  per  minute  .......................      20452. 

Horse-power,  600  -=-  90  ..........................          6.7. 

An  engine  made  by  the  Messrs.  Dow  had  turbine  wheels 
5f  inches  in  diameter  ;  shaft,  %  inch  diameter  ;  depth  of  buck- 
ets, T3T  inch  ;  depth  of  guides  also  T\  inch  ;  weight  of  moving 
parts,  7  pounds,  7  ounces  ;  weight,  including  casing,  68 
pounds  ;  highest  measured  speed  with  70  pounds  steam, 
35000  revolutions  per  minute  ;  so  that  the  velocity  of  the 
circumference  was  nearly  nine  miles  per  minute. 

An  approximate  computation  of  its  regular  daily  perform- 
ance at  70  pounds  pressure  gave  about  8  horse-power  with 
about  75  pounds  of  steam  per  horse-power  per  hour,  the 
speed  being  about  25000  revolutions  per  minute.  Accurate 
measurements  will  doubtless  modify  these  results. 

2.  Required  the  number  of  revolutions  per  minute  neces- 
sary to  burst  a  cast-iron  disk  from  the  centrifugal  force,  the 
modulus  of  tenacity,  T  being  20000  and  the  diameter  of 
the  disk  6  inches,  there  being  a  hole  1  inch  in  diameter  at 
the  centre  for  the  shaft,  weight  of  a  cubic  inch  J  of  a 
pound. 

Assume  the  centre  of  gravity  of  each  half  to  be  at  7-  =  1.3 
inches  from  the  centre. 


*  =  -  30  JX*  ^000  0  XI. 


CHAPTER  Y. 

REFRIGERATION. 

159.  A  refrigerating  machine  is  a  device  for  pro- 
ducing relative  cold.  It  has  been  repeatedly  shown  in  the 
preceding  pages  that  in  any  fluid  doing  work  by  expansion, 
without  transmission  of  heat,  the  temperature  is  lowered. 
Advantage  may  be  taken  of  this  fact  to  produce  a  low  tem- 
perature. Let  m  JV,  Fig.  79,  be  the  volume  of  a  pound  of 
the  fluid  when  the  cylinder  of  a  compressor  is  full ;  let  it  be 
compressed  adiabatically  to  B 
and  at  constant  pressure  to  A  ; 
thence  expanded  adiabatically  to 
J  and  at  constant  pressure  to  C. 
If  the  fluid  be  a  compressible 
gas,  the  temperature  will  de- 
crease from  B  to  A  and  increase 
from  J  to  C ;  but  if  it  be  a 
vapor  the  temperature  will  be  constant  at  constant  pressure 
— some  or  all  of  the  vapor  being  condensed  during  com- 
pression, and  evaporation  taking  place  during  expansion. 
In  both  cases  heat  must  be  abstracted  from  the  working 
fluid  during  the  operation  A  B— the  heat  being  carried 
away  by  the  cooling  substance  ;  and  absorbed  by  the  work- 
ing fluid  during  the  operation  J  C—  being  taken  from  sur- 
rounding substances.  The  latter  result  is  the  one  sought, 
and  is  made  practical  by  placing  the  articles  to  be  chilled  in 
a  room  whose  walls  are  made  practically  impermeable  to 
the  passage  of  heat,  and  abstracting  heat  from  the  room  by 
repeated  operations  like  the  one  just  described,  the  heat  so 


[160.] 


PKACTICAL   OPERATION. 


319 


carried  out  by  the  working  fluid  being  imparted  to  objects 
outside  said  room. 

16O.  Practical  operation.    The  practical  operation 
is  shown  in  Fig.  80,  which  represents  a  vapor  plant.     Omit- 


FIG.  80. 

ting  minor  details,  it  is  as  follows  :  The  working  fluid  is  taken 
into  one  end  of  the  compressor  A  during  the  back  stroke  of 
the  piston,  the  operation  being  represented  by  J  C,  Fig.  79, 
the  volume  of  a  pound  being  m  N  when  the  cylinder  is  full ; 
during  the  forward  stroke  of  the  piston  the  fluid  is  com- 
pressed, the  operation  being  represented  by  C  B,  and  at  B 
the  valve  is  opened  and  the  fluid  forced  into  the  coils  of  the 
condenser  J?,  Fig.  80.  Water  flows  over  the  coils,  reducing 
the  temperature,  if  the  fluid  be  a  permanent  gas,  and  liquefy- 
ing it  if  it  be  a  vapor,  the  operation  being  represented  by 
B  A. 


320  REFRIGERATION.  L1(50-J 

The  heat  absorbed  by  the  water  is  wasted  unless  the 
water  is  used  for  other  purposes.  At  the  left  of  Fig.  80  is 
the  refrigerating  room  C,  which  should  be  enclosed  on  all 
sides,  including  roof  and  floor,  with  several  inches  in  thick- 
ness of  sawdust,  felt,  or  other  non-conductor  of  heat.  This 
room  contains  many  coils  of  pipe  through  which  the  fluid  is 
made  to  flow,  the  coils  being  in  the  centre  of  the  room,  or, 
as  is  often  the  case,  arranged  about  its  walls.  The  fluid 
passes  from  the  condenser  B  to  this  room,  where,  by  properly 
adjusted  cocks,  it  expands  against  a  pressure,  reducing  the 
temperature  and  pressure  until  the  latter  is  that  of  the  initial 
in  the  compressor  ;  the  operation  being  represented  by  A  J, 
Fig.  79.  During  the  back  or  return  stroke  of  the  piston 
the  fluid  flows  into  the  compressor  at  constant  pressure,  the 
pressure  being  maintained  by  the  heat  in  the  refrigerating 
room,  the  operation  being  J  C.  If  the  fluid  be  a  gas,  the 
heat  of  the  refrigerating  room  increases  the  heat  of  the  gas, 
the  temperature  at  J  being  lower  than  that  of  the  room ;  but 
for  a  vapor  the  pressure  and  temperature  are  maintained 
constant  by  the  evaporation  of  the  liquid,  its  volume  being 
increased  from  I)  Jto  D  C. 

It  will  be  seen  that  only  a  part  of  the  changes  here  de- 
scribed are  made  in  the  compressor  ;  however,  the  inaicator 
diagram  C  B  A  J  represents  the  changes  passed  through  by 
the  circulating  fluid,  and  represents  the  work  done  by  the 
compressor. 

Let  the  adiabatics  B  C'and  A  «/be  extended  indefinitely 
to  the  right ;  then  will  the  heat  taken  from  the  working  fluid 
and  carried  away  by  the  condenser  be  G  A  B  F\  and  that 
taken  from  the  refrigerating  room  will  be  G  J  C  F.  The 
operation  is  in  effect  that  of  taking  the  heat  out  of  the  re- 
frigerating room,  adding  heat  to  that  by  the  compressor,  and 
finally  causing  both  heats  to  be  carried  away  by  the  water 
which  passes  through  the  condenser. 

The  operation  of  all  refrigerating  machines  is  essentially 


f161J  EFFICIENCY.  321 

the  same  in  principle :  condense  the  gas  or  vapor,  deprive  it 
of  heat  diminishing  its  volume,  lower  its  temperature  by 
doing  work,  then  expand  it ;  during  the  last  operation  heat 
is  supplied  by  the  articles  to  be  cooled,  and  produces  the  re- 
frigerating effect. 

The  heat  of  the  refrigerating  room  is  carried  out  by  the 
circulating  fluid  to  the  condenser,  where  it  is  carried  away  by 
the  water  of  the  condenser.  During  its  passage  thither  heat 
is  added  to  the  fluid  by  the  work  done  upon  it  by  the  com- 
pressor raising  its  temperature,  and  by  removing  both  heats 
at  the  condenser,  the  circulating  fluid  is  put  into  a  condition 
to  take  up  heat  again  as  it  passes  through  the  refrigerating 
room,  so  that  the  mass  of  circulating  fluid  may  be  constant. 
The  mechanical  operation  of  transferring  the  heat  may  be 
illustrated  by  the  removal  of  water  from  a  chamber  at  a 
lowel  level  than  that  of  surrounding  objects.  For  instance, 
conceive  a  mine  having  springs  of  water,  and  that  the  water 
is  to  be  kept  at  a  low  level ;  or  conceive  a  room  nearly  but 
not  quite  water-tight  submerged  in  a  lake,  and  that  the  water 
in  the  room  is  to  be  kept  at  a  low  level.  By  placing  a  pump 
in  the  room  the  water  may  be  raised,  as  fast  as  it  accumulates, 
to  a  higher  level  than  that  of  surrounding  objects,  from 
which  point  it  will  flow  away  naturally.  If  it  be  not  raised 
sufficiently  high  it  will  not  flow  away.  In  the  refrigerating 
apparatus  the  compressor  raises  the  temperature  of  the  fluid 
to  a  higher  value  than  that  of  surrounding  objects,  thus  en- 
abling the  heat  to  flow  away  ;  and  by  exposing  it  for  a  suffi- 
cient time  it  would  escape  by  radiation  without  the  use  of 
water ;  but  the  condensing  water  hastens  the  process. 

161.  Efficiency.  It  will  be  seen  that  a  refrigerating 
machine  is  a  heat  engine  reversed.  Instead  of  transmuting 
heat  into  work,  work  is  transmuted  into  heat. 

LetN1  =  f^A  G,  Fig.   79,  be  the  heat  carried 
away  by  the  condenser  77,  =  F  C  J  G  the  heat  taken  from 


322  REFRIGERATION.  [ML] 

the  refrigerating  room  and  absorbed  by  the  circulating 
fluid  ;  then  the  work  done  by  the  compressor  upon  the 
fluid  will  be  //,  -  H,. 

The  general  expression  for  the  efficiency  is 

Energy  obtained  (or  work  done) 
Energy  expended 

If  the  energy  obtained  be  the  heat  removed  from  the  re- 
frigerating room,  and  the  energy  expended  be  the  work  done 
on  the  fluid,  then  representing  this  efficiency  by  E^  we  have 

E,  =    jTJ.  (347) 


In  practice  this  will  exceed  unity,  a  result  due  to  the 
peculiar  unit  to  which  the  energy  sought  is  referred.  In 
most  cases  the  energy  obtained  is  a  part  of  the  energy  ex- 
pended, which  is  not  the  case  in  the  above  assumption.  If 
the  energy  obtained  be  referred  to  the  heat  expended,  the 
expression  will  be  less  than  unity.  Thus,  let 

E'  be  the  efficiency  of  the  furnace  compared  with  the 
heat  of  combustion  of  the  fuel, 

E",  the  efficiency  of  the  engine,  compared  with  the  heat 
energy  delivered  to  it  by  the  furnace, 

E'",  the  efficiency  of  the  compressor  referred  to  the  en- 
gine as  unity, 

J5j,  the  efficiency  of  the  refrigerating  system  compared 
with  the  compressor  as  unity  ;  then  will  the  efficiency  of  the 
system  be 

E  =  E'.  E".  E'".  E,.  (348) 

If  the  cycles  were  Carnot's,  and  no  losses  from  clearance, 
friction  or  leakage  in  the  engine  and  compressor,  and  the 
efficiency  of  the  furnace  be  0.70,  then 

E  =  0.70  ^Ll:  .  _!•__,  (349) 


EFFICIENCY.  323 

in  which  r,  is  the  absolute  temperature  of  the  steam  at  the 
furnace,  rt  the  temperature  of  the  refrigerator  of  the  en- 
gine, T4  the  temperature  of  the  condenser,  and  T,  the  tem- 
perature of  the  refrigerating  room.  The  efficiency  realized 
is  far  less  than  this.  It  appears  that  the  efficiency  of  a  re- 
frigerating machine  will  increase  as  the  temperature  of  the 
condenser  decreases,  and  also  as  the  temperature  of  the  re- 
frigerating room,  increases.  This  is  also  apparent  from 
general  considerations,  for  the  higher  the  temperature  of  the 
refrigerating  room  is  allowed  to  be,  the  greater  amount  of 
heat  will  be  carried  away  by  a  pound  of  the  circulating  fluid 
in  expanding  at  constant  pressure,  and  the  lower  the  tem- 
perature of  the  condenser  the  less  the  work  required  of  the 
compressor  in  raising  the  temperature  from  r9  to  r4. 

EXERCISE. 

1.  Let  the  efficiency  of  the  boiler  be  0.75,  of  the  steam 
utilized  by  the  engine  0.15,  of  the  engine  compared  with 
one  without  friction  or  other  waste  0.50,  of  the  compressor 
compared  with  one  without  waste  0.70,  temperature  of  the 
refrigerating  fluid  when  it  leaves  the  condenser  75°  F., 
when  it  leaves  the  refrigerating  room  5°  F.,  and  that  15  per 
cent  of  the  latter  energy  is  lost,  required  the  efficiency  of 
the  plant.  And  if  1  pound  of  coal  fed  to  the  furnace  de- 
velops 12300  thermal  units  'when  completely  burned,  how 
many  pounds  of  ice  at  32°  F.  may  be  formed  from  water 
also  at  32°  F.  for  each  pound  of  coal  burned  ? 

We  have 


E=  0.75  X  0.15  X  0.50  X  0.70  X  =~  X  0.85  =  0.222, 

or  the  efficiency  is  22.3  per  cent ;  that  is,  for  each  thermal 
unit  contained  in  the  coal  fed  to  the  furnace,  0.223  of  a 
thermal  unit  will  be  taken  from  the  refrigerating  room. 


324  REFRIGEKATION.  [162.] 

If  eacli  pound  of  coal  contains  12300  thermal  units,  then 
for  each  pound  burned  there  will  be 

12300  X  0.222  =  2731 

thermal  units  taken  from  the  refrigerating  room,  and  as  144 
thermal  units  are  required  to  congeal  1  pound  of  water  at 
32°  (page  89),  there  may  be  congealed 

2731  ^  144  =  18.96  pounds. 

In  this  solution  it  is  assumed  that  a  Carnot's  cycle  is  per- 
formed. x  If  25  per  cent  of  the  energy  were  lost  instead  of 
15,  the  result  would  have  been  16.8  pounds,  and  this  is  in 
the  vicinity  of  actual  values.  Later  we  will  show  how  purely 
theoretical  results  may  be  found.  If  this  engine  developed 
a  horse-power  with  3£  pounds  of  coal  per  hour,  then  would 
66.40  pounds  of  ice  be  made  per  horse-power  per  hour  from 
water  at  32°  F. 

Compared  with  the  work  done  by  the  compressor  on  the 
circulating  fluid,  the  efficiency  would  be 


=  8.646, 


that  is,  for  every  thermal  unit  of  work  done  by  the  com- 
pressor more  than  5.6  thermal  units  would  be  removed  from 
the  refrigerating  room. 

162.  The  circulating  fluid.  Thermodynamically, 
any  fluid  may  be  the  working  fluid  ;  but  there  are  certain  phy- 
sical and  practical  considerations  which  determine  a  choice. 
It  must  admit  of  a  low  temperature  without  congealing.  Air 
offers  the  advantage  of  being  abundant,  without  cost,  and 
admitting  of  any  desired  range  of  temperature  ;  but  its  den- 
sity being  small,  the  required  apparatus  must  be  correspond- 
ingly large.  If  vapors  are  used  they  must  be  capable  of 
vaporizing  at  low  temperatures.  Among  the  substances 


[163.]  SOME   PROPERTIES   OF   AMMONIA.  325 

used  are  ammonia,  JV  773,  sulphur  dioxide,  $,  <9,  methylic 
ether,  C,  7/6  <9,  and  sulphuric  ether  ;  the  first  two  of  which 
are  the  most  common,  and  of  these  we  will  consider  ammo- 
nia especially.  The  general  formulas  will  be  applicable  to 
any  vapor. 

Generally  brine — water  thoroughly  saturated  with  salt — 
circulates  in  the  coils,  the  brine  being  cooled  in  a  tank  by 
the  ammonia,  as  above  described.  This  saves  a  large  amount 
of  ammonia.  Brine  may  be  produced  that  will  not  congeal 
until  the  temperature  is  below  zero  Fahrenheit. 

163.  Some  properties  of  ammonia.  Certain 
properties  of  ammonia  have  been  determined  by  Regnault, 
but  his  determination  of  the  latent  heat  of  vaporization  and 
the  specific  heat  of  liquefied  ammonia  were  lost  during  the 
reign  of  the  Commune,  in  1870  ;  and  these  we  will  deter- 
mine by  computation  founded  on  the  results  of  experiment 
and  certain  thermodynamic  principles. 

In  delation  des  Experiences,  Vol.  II.,  pp.  598-607,  are 
the  results  of  Regnault' s  experiments  upon  temperature  and 
corresponding  pressure  of  saturated  ammonia.  These  we 
have  plotted  in  Fig.  81,  the  ordinates  to  the  dots  represent- 
ing the  pressures,  and  the  abscissas,  temperatures.  If  the 
law  be  represented  by  Rankine's  formula,  equation  (80),  p. 
97,  the  value  of  C  will  be  so  small  that  its  effect  will  be  in- 
appreciable, and  the  formula 

2196 
com.  logp  =  8.4079  -   -p- ;  (350) 

>r,  if  p  be  pounds  per  square  inch, 

2196 
foffnP  =  6.2495  -  — ^— 

represents  the  results  of  the  experiments  with  much  accu- 


326  REFRIGERATION.  [163.] 

racy  from  about  —  20°  F.  to  100°  F.,  or  from  about  18 
pounds  per  square  inch  to  215  pounds.* 

640 


-40-20     0     20    40    60    80  100120140160180 
Temperatures,  Degrees  Fahr. 
FIG.  81. 

The  specific  heat  of  ammonia  gas  is  0.50836,  which  is  a 
little  more  than  for  steam  (Rel.  des  Exp.,  II.,  p.  162). 
Density  of  lique/ed  ammonia,  that  of  water  being  unity. 

*  In  the  Transactions  of  the  American  Society  of  Mechanical  Engineers 

OOftA 

for  1889,  I  used  the  formula  log  p  =  6.2469  -  — '  and  showed  the  dif- 
ference between  the  computed  and  observed  values.  This  formula  is 
nearer  correct  for  higher  pressures. 


[163.]  SOME  PROPERTIES  OF  AMMONIA.  327 

Temp.  Density.  Dif.  Authority. 

At       15.5°  C.... 0.731 Faraday. 

-10         ....0.6492        1 

-63 
-    5         ....0.6429 

—65 


0         ....0.6364 


D'Andreeff:    An.      (3),     56,     317 


("  Smithsonian  Miscellaneous  Col- 
. . .  .0.6298^^  y     lections,"  Vol.  XXXII.,  1888). 

10         ....0.6230 

—70 
15          0.6160 

—71 


J 

These  may  be  expressed  very  nearly  by  the  formula 
3  —  0.6364:  -  0.0014  t 

=  0.6502  -  0.000777  r, 

when  t  is  degrees  centigrade  and  T  degrees  Fahrenheit. 

Density  of  the  gas. — Regnault  gives,  for  the  theoretical 
density  of  the  gas,  0.5894  (Bd.  des  Exp.,  Yol.  II.,  p.  162), 
but  he  also  says  :  "  The  real  density  of  ammonia  gas  is  cer- 
tainly higher  than  the  theoretical;  the  only  experimental 
density  of  which  I  have  knowledge  gives  0.596  "  (Ibid., 
Yol.  III.,  p.  193).  We  will  use  the  latter  value. 

Volume  of  abound  of  the  gas  at  the  melting-point  of  ice. 
We  have 

Weight  of  litre  of  air  at  0°  C.,  760m 1.293187  grammes, 

or  weight  of  cu.  metre  of  air  at  0°  C.,  760m..1.293187  kilog. 

(Ibid.,  Yol.  L,  p.  162.)     Hence  the  weight  of  one  litre  of 
the  gas  at  0°  C.,  760""%  will  be 

1.293187  X  0.596  =  0.770739  grammes, 
and  the  volume  of  one  gramme  of  the  gas  will  be 

0^739= 


328  REFRIGERATION.  [164.  , 

Reducing  this  to  the  equivalent  of  one  pound  and  cubic- 
feet  gives 

35.3161 
1.2973  22(M6   =  20.7985  cu.  ft.  per  Ib.  =  vf 

Value  of  R. 

&  r0  _  2116.3  X  20.7985  X  _ 
~^~  492.66 

This  is  89.343  -=-   778   =    0.11483  of  a  thermal  unit; 
•  hence,  at  this  state,  equation  (28),  p.  49, 

£¥  =  0.50836  -  0.11483  =  0.39352  ;  (353) 

and,  equation  (31) 

y  =  1.292;  (354) 

and.  although  y  will  not  be  constant,  it  will  practically  be 
so  for  the  superheated  gas. 

164.  To  find  the  latent  heat  of  evaporation 

of  Ammonia.     From  equation  (74),  p.  95, 


in  which  v,  is  the  volume  of  a  pound  of  the  liquid  ;  and  as 
this  is  small  compared  with  the  volume  of  a  pound  of  the 
vapor  it  may  be  omitted,  and  we  have,  omitting  also  the 
subscripts, 

ke  =  rv  £j~  -T-  778.  (355) 

From  equation  (350)  we  have 

^2.  =  2196  X  2.3G26  2L  • 

.  •  .  A.  =  6.49922  ^.  (356) 

At  the  state  when 

PJL  =  ?!_H?  =  80  343,  we  have 

T  r0 

A.  =  580.66. 


[164.; 


LATENT  HEAT   OF   EVAPORATION. 


329 


This  result  must  be  for  a  state  where  v  >  ea  ;  for  the  general  theory  of 
imperfect  gases  shows  that  for  the  same  volume  p  +  r  is  less  for  a  small- 
er pressure,  and  in  this  case  at  the  pressure  pa  the  gas  is  superheated, 
and  at  the  point  of  saturation  p  will  be  less  than  p0 ;  hence  tfie  latent 
heat  of  evaporation  of  ammonia  mmt  be  less  than  580.66  when  the  specific 
volume  is  20.7985  cubic  feet* 

The  general  value  of  p  v  +  T  will  be  found  from  the  equation  of  the 
gas.  In  Vol.  II.  of  Expei-iences,  p.  152,  Regnault  has  given  the  results 
of  his  experiments  upon  the  elastic  resistance  of  ammonia  at  the  constant 
temperature  of  8.1°  C.  (46.58°  F.).  These 
give  the  relations  between  the  pressures 
and  volumes  of  the  actual  isothermal  A  C, 
Fig.  82  ;  the  isothermal  of  the  gas  pass- 
ing through  A,  if  perfect,  being  A  B. 
These  experiments  reduced  to  volumes  in 
cubic  feet  per  pound,  and  pressures  in 
pounds  per  square  foot,  are  given  in  the 
following  table  : 


\ 


FIG.    82. 


TABLE. 


RELATIONS  BETWEEN  VOLUMES  AND  PRESSURES  OF  AMMONIA  GAS  AT 

THE   TEMPERATURE   46.58°  F.f 


V  olumes  cu.  ft.  per  ID. 

Lbs.  per  sq.  ft. 

Lbs.  per  sq.  in. 

24.3716 

1862.706 

12.93 

23.157 

1958  976 

13.60 

21.944 

2064.096 

14.33 

20.7985 

2178.960 

15.13 

19.563 

2311.200 

16.05 

18.365 

2458  800 

17.08 

17  160 

2618.784 

18.19 

15.961 

2822.544 

19.60 

14.762 

3042.288 

21.13 

13.557 

3303.648 

22.94 

12.355 

3617.768 

2512 

11.1412 

3996.820 

27.76 

*  This  shows  that  the  latent  heat  found  by  Ledoux  is  erroneous  (Ice-Makimj  Machines, 
by  M.  Ledoux,  Van  Nostrand's  Science  Series,  No.  40).  For  the  volume  20.8  cubic  feet 
Ledoux  gives  about  600  B.  T.  U. 

t  Trans.  Am.  Soc.  Mech.  Engineers,  1889. 


330  KEFKIGEKATION.  [164.] 

Assuming  in  equation  (4),  p.  13,  ac  =  o,  and  neglecting   all    terms 
after  «,,  it  may  be  written  in  the  form 


The  first,  sixth,  and  last  experiments  of  the  preceding  table  give  for  the 
products  p  v,  and  the  corresponding  values  v, 


r  (24.3716)" 
b 


=  pv  =  45397. 
=  v  T  =  45156. 


r  (18.365)° 

=  44529. 


r(l  1.141)" 
In  these  equations  r  =  507.24,  and  they  give 

a  =  91  005,  b  =  16921  r,  n  =  0.97. 

Letting          a  s  91,  b  =  16920  r,  n  =  0.97. 

7%€  equation  of  the  gas  will  be 


and  hence,  equation  (356), 

The  latent  heat  of  ammonia  is     . 

5065.7  16920 


=  592.52(1-1^*)  (358) 

We  now  proceed  to  find  the  latent  heat  for  certain  states  of  the  fluid. 
In  Fig.  83,  a  represents  the  state  of  ammonia  gas  at  the  temperature  of 
melting  ice  under  the  pressure  of  one  atmosphere,  for  which  the  volume, 
as  found  above,  is  20.7985  cubic  feet  per  pound  ;  that  is, 

o  h  =  20.7985,  h  a  =  2116.3,  r  =  492.66. 

Let  state  «  represent  the  pressure  and  volume  of  the  first  experiment  in 
the  preceding  table,  for  which 

o  t  =  24.3716,  t  s  =  1862.7  Ibs.  per  sq.  ft 

State  e  is  the  last  in  the  table,  for  which 

oj  =  11.141,  je  = 


[164.] 


LATENT   HEAT   OF   EVAPORATION. 


331 


From  s  to  e  is  the  actual  isothermal  of  the  gas  as  determined  by  Reg- 
nault's  experiments  for  the 
temperature  of  46.68°  F. 
As  this  was  the  tempera- 
ture of  the  water  sur- 
rounding the  tube,  the 
temperature  of  the  gas 
may  have  been  somewhat 
less  ;  but  we  use  this  value 
as  exact.  The  isothermal 
prolonged  intersects  the 
curve  of  saturation  in 
m. 

To  find  the  latent  heat 
of  evaporation  at  c,  having 
the  same  volume  as  at  a, 
it  is  necessary  to  find  the 
temperature  at  this  point. 
Make 

v  =  20.7985 
and  equation  (357)  gives 


_       91 
P  ~  20.7985 


16920 


(20.7985) 

=  4.3753  r  -  42.843  ; 
which  substituted  in  equation  (350)  gives 

log  (4.3753  T  —  42.843)  =  8.4079  -  *—  ; 

Eq.  (350) 

and  equation  (358)  gives 

A.  =  578.96. 

It  was  found  above  for  this  volume,  20.7985,  that  if  p  .«.  T  were  con- 
stant down  to  the  point  of  saturation,  the  latent  heat  would  be  580.66,  a 
value  exceeding  that  just  found  by  only  1.70  thermal  units— a  difference 
in  the  right  direction,  and  of  reasonable  amount.  This  test  being  satis- 
factory, we  now  apply  it  to  other  states. 


.  '  .p  =•  1823.7  Ibs.  per  sq.  foot. 
=  12.7  Ibs.  per  sq.  inch  ; 


332  REFRIGERATION.  f!64.] 

For  the  state  immediately  below  «,  on  the  curve  of  saturation,  we 
have 

v  =  24.372, 

and  with  the  same  equations  as  in  the  preceding  case,  there  results 

T=    420.4;  .  '.  1  =—  40°.26F., 
p  =  1531.1  Ibs.  per  sq.  ft.  =  10.6  per  sq.  inch, 
he  =  579.67  thermal  units. 

For  state  m, 

r  =  507.24,  or  r=468.58F., 
and  from  the  same  equations 

p  =  11988  per  sq.  ft.  =  83.25  per  sq.  inch, 
v  =  3.41  cubic  feet, 
he  =  526.47  thermal  units. 

For  the  state  for  which 


we  find 

r  =  4687;  .-.  T=  8M  F., 
p  =  5279  per  ft,  =  36.8  per  inch, 
Ae  =  550.52. 

Assuming  the  form  of  expression  adopted  by  Regnault  for  the  latent 
heat  of  evaporation  of  all  substances, 

he=d-eT-fT*,  (359) 

and,  using  in  it  the  data  for  the  three  last  cases  just  given,  we  have 

526.47  =  d  -  46  .58  e  -  2169  7  /, 
550.52  =  d  -  8.1  e  -  6561  /, 
579.67  =  d  +  40  e  —  1600  /. 

These  give 

d  =  555.50,  e  =  0.61302,  /  =  0.000219, 

and  equation  (359)  gives  the  following  as 

A  more  convenient  formula  for  the  latent  heat  of  evapo- 
ration of  ammonia  : 

he  =  555.5  -  0.613  T  -  0.000219  T.  (360) 

The  latent  heat  of  ammonia  vapor  in  the  table  at  the  end 
of  the  volume  has  been  computed  by  means  of  this  formula. 


[165,  166,  167.]  AMMONIA   VAPOR.  333 

165.  Specific  volume  of  liquefied  ammonia. 

If  the  volume  of  a  pound  of  water  be  0.016  of  a  cubic  foot, 
then  will  the  volume  of  a  pound  of  liquid  ammonia  be, 
equation  (351), 

*,   =  -          °-016  (361) 

0.6502  -  0.000|  T 

This  formula  is  sufficiently  accurate  for  temperatures  be- 
tween —  5°  F.  and  100°  F.  A  mean  value  gives  about  41 
pounds  per  cubic  foot. 

166.  Specific  volume  of  ammonia  gas.    From 

equation  (84),  page  98,  ' 


By  the  aid  of  equation  (350),  after  omitting  the  sub- 
script s,  we  have 

*  =  63558  •!  +  ""  (362) 

The  volumes  in  the  table  of  the  Properties  of  Saturated 
Ammonia  were  computed  from  this  equation.  Since  vl  is 
small  compared  with  -y,  it  may  generally  be  omitted. 

167.  Isothermals  of  ammonia  vapor.    If  the 

vapor  be  saturated,  the  isothermal  will  be  parallel  to  the  axis 
of  v,  as  A  B,  Fig.  74. 

If  the  vapor  be  superheated,  the  equation  will  be  (357), 
after  making  r  constant.  It  will  be 

16920  ,oftoN 

V  v  =  91  r.  —      „„,     .  (363) 

v 

The  general  equation  of  vapors  in  which  the  last  term  is 
a  function  of  v  only,  will  be 

pv  =  ar-*-.  (364) 

if  v* 


334  KEFKIGERATION. 

168.  Adiabatics  of  ammonia  vapor.     If   the 

vapor  be  continually  saturated,  the  equation  of  the  adiabatic 
will  be  (a)  or  (J),  page  184,  or 

u  =  x  v  =  (o  log      +  ^AL   T--,  (365) 


in  which  u  is  the  volume  of  the  vapor  and  liquid  when  only 
the  07th  part  of  the  liquid  is  vaporized  ;  but  as,  in  our  analy- 
sis, the  volume  of  the  liquid  compared  with  the  vapor  is 
neglected,  it  really  represents  the  volume  of  the  orth  part  of 
a  pound  of  vapor  ;  c  is  the  specific  heat  of  liquid  ammonia, 
the  experimental  value  of  which  is 

c  =  1.22876. 

If  the  vapor  le  superheated,  then  the  first  of  equations 
(A),  page  48,  and  equation  (364)  give 

dH=Kvdr  +  rt  d  v. 

v 

But  for  an  adiabatic 


d  t  dv 

—  =  —  a  — 

r  a  v 


T, 

where  vt  and  T,  are  inferior  limits,  and 


(366) 


To  obtain  an  equation  between  p  and  v  eliminate  T  between 
(366)  and  (364),  giving 


[169,  170.]  LIQUID   AMMONIA.  335 

For  ammonia  gas,  these  become 


,  =  911- (*)"""_"?»  ( 

v9  \v  /  v 

(371) 
the  last  of  which  is  in  terms  of  p  and  r  as  variables. 

169.  The    specific  heat  of  the  saturated  vapor  of 
ammonia  of  constant  weight  is  negative.     Equation  (139), 
page  147,  gives,  omitting  terms  containing  r\ 

837.5 
s  =  c . 

T 

If  c  =  1,  this  will  be  negative  for  values  of  T  less  than  837°, 
or  377°  F. ;  hence,  for  the  range  of  temperatures  ordinarily 
used  in  engineering  practice,  the  specific  heat  of  saturated 
ammonia  is  negative,  and  the  saturated  vapor  will  condense 
with  adiabatic  expansion,  and  the  liquid  will  evaporate  with 
the  compression  of  the  vapor,  and  when  all  is  vaporized  will 
superheat. 

Thus,  in  Fig.  84,  if  B  Cs  be  the  curve  of  saturation,  and 
the  vapor  be  compressed  adiabatically  from  any  point,  as  (7, 
on  the  curve  of  saturation,  the  adiabatic  C I  will  rise  above 
B  C,  and  if  it  be  expanded  from  the  same  point  it  will  fall, 
below  Cs.  Equation  (370)  is  the  equation  of  C  I,  and 
(365)  of  CK,  the  part  below  C. 

170.  Specific  heat  of  liquid  ammonia.      As- 
sume the  volume  m  M,  Fig.  84,  of  the  pound  of  liquid  to  be 
constant  at  all  pressures,  and  let  M  D  be  the  absolute  pres- 


336  KEFRIUKRATION.  [170.] 

sure  at  the  absolute  temperature  T,  B  Cs,  the  curve  of  satu- 
ration, D  II,  A  G,  B  F,  IK,  adiabatics. 
Let  the  vapor  be  expanded  from  D  at  the 
pressure  p  and  temperature  T  until  it  is 
ail  evaporated  at  state  C,  thence  com- 
pressed adiabatically  to  /,  thence  com- 
FIG.  84.  pressed  at  constant  pressure  to  A,  where 

it  is  liquefied,  thence  by  the  abstraction  of  heat  let  the  pres- 

sure be  reduced  to  D  ;  then 

II  DAG-}-  GAIK=  IIDCK+  DCIA. 

Let  the  temperature  of  A  B  be  T  -j-  d  r,  and  of  I,r-\-d  T\ 
for  the  vapor  from  B  to  I  will  be  superheated,  its  tempera- 
ture increasing  with  increase  of  volume  ;  then,  if  c  be  the 
specific  heat  of  the  liquid, 

HD  A  G  =  Jcdr, 
DCIA  =  vdp, 


G  A  IK  =  GABF+FB1K, 


Equation  (360)  gives,  since  d  T  =  d  T, 

-  (j^  =  0.6130  +•  0.000438  T, 
From  equation  (350)  find 


dP  =  6.49922^. 
dr  T* 


. 
J 

Differentiating  (371),  after  which  change  dr  to  dr'  and 
drop  all  subscripts  in  the  second  member,  and  (372)  becomes 

.-«.«  +  0.000438  T+ 


[171.]  COMPRESSOR   AND   CONDENSER.  337 

As  this  investigation  depends  upon  a  comparatively  small 
range  of  volumes,  from  11  to  24  (table  011  page  329),  and 
assumptions  in  regard  to  the  forms  of  the  functions  in  equa- 
tions (350)  and  the  first  equation  on  page  330,  it  may  not  be 
very  reliable  for  a  large  range  of  temperatures.  We  will 
compute  a  value  for  a  volume  within  the  limits  of  the  table, 
and  will  take  values  given  on  page  331,  viz. : 

v  =  20.7985,  T  =  426.6,  T  =  -  34°  ¥.,p  =  1823.7  Ibs., 
for  which  we  find 

c  =  1.093. 

The  mean  of  eight  determinations  made  by  Dr.  Hans  von 
Strombeck,  chemist,  gave 

c  =  1.22876. 

The  experimental  value  was  determined  when  the  tem- 
perature of  the  liquid  was  about  80°  F. 

KEFRIGEKATING    SYSTEM  WITH  VAPOR   CONTINUALLY   SATU- 
RATED. 

171.  Work  of  the  compressor  and  condenser. 

In  Fig.  85  let  E  F  be  any  adiabatic,  p»  r,  apply  to  A  -B, 
p^  r2  to  D  C,  a?,  the  fraction  of  a 
pound  of  vapor  at  state  _£*,  x^  at    A 
Fj  sc3  at  A  (when  it  is  not  zero), 
«J4  at  J,   vl  the  volume  of  a  pound    G 
of  vapor  at  B,  v^  the  volume  of  a 
pound  under  the  pressure  j?a ;  the 
odd  figures  for  subscripts  belong- 
ing  to   the   upper  line,  and   the  FIG.  g5. 
even  to  the  lower. 

The  work  of  compression  from  F  to  E,  and  of  forcing  it 
into  the  condenser  from  Eio  A,  will  be,  equation  (m\  page 
192, 

U'=  A  EFD  =  J  \c  (T,-  rt)  -f 


338  REFRIGERATION.  [1~1-J 

In  the  expansion  chamber  the  vapor  expands  against  a  re- 
sistance, reducing  the  pressure  from  pl  to  p^  and  tempera- 
ture from  rt  to  r,,  doing  the  work  A  D  J,  and,  assuming 
an  adiabatic  change,  the  expression  for  the  work  will  be 
found  by  writing  xt  for  a?,  and  a%  for  a?,  in  the  preceding 
equation,  since  all  the  other  quantities  will  remain  the  same  ; 

(375) 

After  the  vapor  is  forced  into  the  condensing  chamber,  its 
specific  volume  is  reduced  by  condensation  from  A  Eto  A 
(and  for  the  sake  of  generalizing  the  analysis  we  assume  for 
the  present  that  it  is  not  reduced  entirely  to  a  liquid  at  A, 
but  that  a%  has  a  finite  value),  and  the  heat  emitted  from  the 
circulating  fluid  —  and  absorbed  by  the  condenser  —  will  be 
the  area  between  EA  and  the  indefinitely  extended  adiabat- 
icsA  J  m&EF,  or, 


and  the  heat  absorbed  by  the  circulating  fluid  (ammonia  or 
brine),  which  is  taken  from  the  refrigerating  room,  will  be 

-7/,  =  -e/te-<>AM;  (376) 

then, 

U  -  U"  =  K>  -  //,. 

Since  the  cycle  is  Carnot's,  we  have 

fo  -  a?,)  Ae.      fa  -ap  Aef  . 


The  efficiency,  referred  to  the  work  done  by  the  com 
pressor,  will  be 

(878) 


[171.]  COMPRESSOR   AND   CONDENSER.  339 

which  is  independent  of  the  amount  of  liquid  evaporated, 
as  it  should  be.  More  heat,  however,  will  be  removed  by 
the  work  of  the  compressor  if  all  the  liquid  be  evaporated 
that  is  possible  and  leave  it  continually  saturated  ;  that  is, 
if  the  fluid  be  vapor  only  at  B  ;  and  also  if  it  be  entirely 
condensed  at  A.  These  conditions  require  that  a?,  =  1  and 
x3  =  0  ;  and  the  equations  for  largest  effect  become 


U1  =  J    c>  (TI  -  rt)  +  Jfll  -  x,  /.  (379) 

V"  =  J^c(rl-T^-x<h^.  (380) 

HI  =    Jhn.  (381) 

//a  =  J  (a>,  -  a>4)  Aea.  (382) 

H,  -  Hy  =  J  \~hn  -  (a>,  -  x,}  Ae,l  =  V  -  U".(3S3} 


Eq.  (365),   aj,  =.  (  c  %e  ^  +  ^J  £ .  (384) 

For  a?,  =  0,  aj4  =  c   -p-  %e  —  •  (385) 

"ea  7a 

=  2.3026  c  ^-  fc<7..  — . 


These  formulas  solve  the  theoretical  part  of  the  problem. 
The  temperature  of  the  circulating  fluid  in  the  condenser 
will  be  several  degrees  above  that  of  the  condensing 
water,  depending  upon  the  amount  of  condensing  surface — 
say  about  10°  F.  above ;  and  the  temperature  of  the  con- 
densing water  will  depend  upon  external  circumstances, 
and  may  be  40°  F.  in  winter  and  60°  F;  to  70°  F.  jn  sum- 
mer. Also  the  temperature  of  the  refrigerating  room  will 
be  higher  than  that  of  the  circulating  fluid— say  some  10°  F. 
— depending  upon  the  rapidity  of  the  circulation  and  the 
amount  of  surface  of  the  pipes.  For  the  manufacture  of 
ice,  the  refrigerating  room  is  kept  at  a  temperature  of 


340  REFRIGERATION.  1172,  173,  174.] 

about  15°  F.,  and  that  of  the  brine  may  be  between  0°  F. 
and  5°  F. 

172.  Volume  of  the    compressor    cylinder 

per  n  pounds  of  ammonia  per  stroke. 

Let  V  be  the  required  volume  in  cubic  feet  and  v  the 
volume  of  a  pound  of  ammonia  gas  at  the  lower  tempera 
tu re,  equation  (3b'2),  then 

V  =  n  v.  (386) 

173.  Volume    of   the   compressor  cylinder 

to  produce  a  given  refrigerating  iffect. 

Let  A9  be  the  thermal  units  abstracted  per  pound  of 
ammonia, 

q  =  nji^  the  required  number  of  thermal  units  to  be  ab- 
stracted—  which  will  also  be  a  measure  of  the 
refrigerating  power, 

V,  the  volume  swept  over  by  the  piston  or  pistons — con- 
sidered single-acting — per  revolution, 

-V,  the  number  of  revolutions  per  minute, 

v,  the  volume  of  a  pound  of  ammonia  gas  at  the  lower 

temperature, 
then 

N  V 


.  (3'7) 

174.  Duty.  The  duty  of  a  refrigerating  plant  nuiv 
be  referred  to  the  number  of  thermal  units  required  to  UK  h 
one  pound  of  ice.  The  latent  heat  of  fusion  of  ice  at  the 
pressure  of  one  atmosphere  is  144  thermal  units  (page  89), 
and  if  A,  be  the  thermal  units  abstracted  by  the  circulating 
fluid  per  pound,  then  will  the  duty  be 

Ice-capacity,  Ibs.  =  -A-.  (388) 

J.TT 


[174.]  DUTY.  341 

If  referred  to  one  pound  of  coal  fed  to  the  furnace,  then 


Pounds  of] 
circulating 
fluid  pei- 


Specific 
heat 


uuiu  pci  I 

3ity  _    _  hour  _ 
per  Ib.  of  "fuel,  Ibs.  I44"x  pounds  of  fuel  per 


Range  of 


tempera- 
Lure 


If  w'  pounds  of  coal  evaporates  W  pounds  of  steam  per 
hour,  then 

Pounds  of  steam  per  Ib.  of  fuel  —  (390) 

w 
which  may  be  used  in  equation  (389). 

EXERCISES 

1.  How  many  thermal  units  will  be  removed  from  the 
refrigerating   room   per   pound   of   ammonia,   the   highest 
average  temperature  of  the  liquid  being  62°   F.,  and  that 
after  expansion  —  1°  F.,  the  vapor  being  continually  satu- 
rated, and  the  specific  heat  of  the  liquid  1.06  ? 

Omit  0.66  in  the  absolute  temperature. 

FromEq.  (360),  ^  =  556.1. 

"       "    (384),  (1.06X  2.3026%,  g  +  ^)^-, *  =  0.9808. 

\  40"  0*-*    /    000. 1 

"       "     (385),  1.06  X  2.3026  -^-  log™  !j|?        x<  =  0.1125. 

oob.  1  4o» 

"       "     (382),  7<3  =  455.16. 

That  is,  each  pound  of  ammonia  will  carry  away  455.06  thermal  units. 
There  must  be  taken  into  the  compressor  j-a  =  93.08  per  cent  of  vapor, 
and  therefore  6.92  per  cent  of  liquid.  There  must  be  xt  =  11.25  per 
cent  of  liquid  vaporized  during  the  rise  of  temperature  from  —1°  F.  to 
62°  F. 

2.  In  the  preceding  exercise  if  the  ammonia  cools  a  brine 
whose  specific  heat  is  0.8,  the  lowest  temperature  of  the 
brine  being  8°  F.,  the  highest  15°  F.,  how  many  thermal 
units  will  be  removed  by  the  brine  per  pound,  and  how 
many  pounds  of  brine  will  be  required  to  absorb  the  heat  of 
one  pound  of  the  ammonia  ? 


342  REFRIGERATION.  [174.] 

The  heat  imparted  to  the  brine  raises  its  temperature  15  —  8  =  7"  F.  ; 
hence  each  pound  will  absorb  0.8  X  7  =  5.6  thermal  units. 
The  ammonia  imparts  to  the  brine,  equation  (382), 

/,„  =  (j-,  -  set)  556.1  =  454.06. 

Let  z  be  the  pounds  of  brine  necessary  to  absorb  this  heat  in  raisini:  its 
temperature  one  degree,  then 

0.8  X  7  z  =  454.06  ; 

.  •  .  z  =  81.17  pounds. 

When  brine  is  used,  larger  pipes  or  greater  velocity  of  the  brine  will  be 
required  than  if  ammonia  only  were  used,  and  a  lower  ti'iuperature  <>f 
the  ammonia  will  be  required. 

3.  What  will  be  the  efficiency  in  Exercise  1  i 
Equation  (378)  gives 

E  _  459_72« 
El~   63~~ 

That  is,  for  every  thermal  unit  of  work  dorw  by  tlie  comprtwnr.  7.2S 
thermal  unit*  will  be  removed  from  the  cold  room.  If  the  work  done  by 
the  compressor  be  expressed  in  foot-pounds,  and  the  heat  removed  be  in 
thermal  units,  then 

E,  =  ~  =  0.0093. 

That  is,  for  every  foot-pound  of  work  done  by  the  compressor.  0.0093  of 
a  thermal  unit  will  be  abstracted  from  the  cold  room. 

4.  How  many  pounds  of  water  will  be  required  by  the 
condenser  per  pound  of  ammonia  if  condensed  to  a  liquid 
at  63°  FM  the  temperature  of  the  water  being  increased 
from  50°  F.  to  64°  F.  ? 

The  heat  surrendered  by  the  ammonia  will  be7<i  =  h,t,  equation  (381); 
and  since  the  specific  heat  of  water  is  unity,  we  have 

y  =  -~  =  36.9  pounds. 

5.  How  many  cubic  feet  of  water  will  be  required  in  the 
preceding  exercise,  per  pound  of  ammonia  ? 

If  less  water  is  desirable,  a  greater  range  in  its  temperature  must  bo 
allowed,  and  if  a  less  range  of  temperature,  then  more  water  will  be 
required. 


[174.]  DUTY.  343 

G.  What  must  be  the  volume  of  a  single-acting  compres- 
sor cylinder  in  Exercise  1  that  it  shall  compress  ^  of  a  pound 
of  ammonia  gas  with  each  stroke  ?  Equation  (386). 

7.  In  Exercise  I  find  the  highest  and  lowest  pressures  in 
the  compressor.     Equation  (350). 

8.  In  Exercise  1  find  the  greatest  and  least  specific  vol- 
urnss  of  the  gas.     Equation  (362). 

9.  In  Exercise  1  find  the  external  latent  heat  of  evapora- 
tion at  the  higher  and  lower  temperatures.     (This  will  be 
pl  v)  and  j\  v.2 —  neglecting  the  volume  of  the  liquid.) 

10.  Find  the  negative  work  done  by  the  compressor  dur- 
ing one  revolution  and  the  positive  work  done  by  the  one 
pound  of  ammonia  in  Exercise  1. 

Negative  work,  Eq.  (379),  IT  =  856577. 

Positive  work,  Eq.  (380),  U"  =  3282. 

11.  If  a  compressor  have  two  cylinders  of  equal  size,  each 
single-acting,  diameter  of  each  piston  18  inches,  stroke  2± 
inches,  ammonia  gas  entering  at  0°  F.  and  forced  out  at  629 
F.,  each  32  revolutions  per  minute  ;  how  many  thermal  units 
will  be  removed   from  the  cold  room,  no  allowance  being 
made  for  losses  and  the  ammonia  gas  continually  saturated  ; 
what   the   indicated   horse-power   of    the   compressor,    the 
specific  heat  of  liquid  ammonia  being  1.08  ? 

Also,  if  15  per  cent  be  lost  on  account  of  clearance  and 
imperfect  working  of  the  compressor  and  50  per  cent  of  the 
remainder  due  to  losses  in  the  refrigerating  room  ;  how 
many  pounds  of  ice  may  be  made  from  water  at  62°  F. 
cooled  to  32°  and  frozen  and  the  ice  cooled  to  2-t°  F.,  the 
specific  heat  of  the  water  being  unity  and  of  the  ice  0.50? 

Revolutions  per  hour n  =  1920. 

Volume  of  each  cylinder  it  X  81  X  24  *  1728  cu.  ft..  V  —  3.533. 

Working  vol.  displaced  by  pistons  per  hour,  cu.  ft. . .  2  n  V=  13567. 

Specific  vol.  per  Ib.  of  gas  at  0°  F.,  Eq.  (362),  cu.  ft. .  e  =  9.0. 

Weight  of  gas  compressed  per  hour,  2  n  V  •*•  v,  Ibs. .  W  =  1507. 

Vapor  taken  in  per  pound  of  fluid,  Eq.  (384) x,  ~  0.932. 


344  KEFKIGKUATIOX.  U?5.] 

Liquid  evaporated  to  reduce  temperature,  Eq.  (385).. .  .*•«  =  0.111. 

Latent  lieat  of  evaporation  at  0°  F <  h&  =  555.5. 

Refrigeration  per  pound,  Eq.  (382),  thermal  units ht  =  456.1. 

Condenser,  heat  removed,  Eq.  (381) /ti  —  517.5. 

Work  per  pound,  thermal  units (hi  —  ?h)  =  61.4. 

"  minute,  thermal  units (/;,  -  7ta)  W  *  60  =  1542.3. 

Horse-power,  1542.3  -  (33000  -  778) //.  P.  =  36. 36. 

Efficiency,  Eq.  (378) Et  -  7.42. 

Heat  removed  from  cold  room  per  hpur W  h*  =  687400. 

Effectual  heat  removed,  0.85  X  0.50  X  687400 =  292145. 

Heat  expended  in  freezing,  per  lb.,  30  -f  \  of  8  -f  144,  =  178. 

Ice  per  hour,  292245  -  178,  Ibs =  1641. 

Ice  per  horse-power  per  hour,  Ibs =  45.1. 

The  IHP.  of  the  steam-engine  should  be  about 

36.36*0.6 H.P.K  60. 

Ice  per  IHP.  would  then  be  45.1  X  0.6  Ibs 27. 

At  4  Ibs.  coal  per  IHP.  per  hour,  Ibs.  of  ice  per  lb. 

coal =        6.8. 

12.  If  120000  pounds  of  brine  passing  through  the  cold 
room  per  hour  has  its  temperature  increased  5.2°  F.,  spe- 
cific heat  0.80,  at  an  expenditure  of  20(M)  pounds  of  steam 
generated  by  the  burning  of  200  pounds  of  coal ;  how  many 
pounds  of  water  may  be  frozen  at  and  from  32°  F.  per  pound 
of  coal  burned?  Ans.  17. 

(This  data  is  almost  exactly  that  of  an  actual  case.) 

CASE  IN  WHICH  THE  GAS  is  SUPERHEATED  BY  COMPRESSION. 

175.  Superheating.  It  will  be  seen  from  the  pre- 
ceding exercise  that  the  adjustment  of  liquid  to  vapor,  in 
order  to  insure  the  largest  result  per  pound  of  ammonia 
must  be  delicate,  and  can  hardly  be 
realized  in  practice ;  we  therefore  will 
now  assume  that,  when  the  compressor 
cylinder  is  full,  the  fluid  is  all  gas,  with  no 
liquid  present,  and  is  superheated  at  state 
C.  In  compressing  it  adiabatically,  let 
the  path  be  C  I.  If  C  were  on  the  curve  of  saturation,  the 
vapor  would  become  superheated  by  adiabatic  compression. 


t175J-  SUPERHEATING.  345 

Let  the  state  /  be  designated  by  accents,  as  v',  p',  rf ; 
and  C  by  double  accents,  as  v",  p",  r" . 

The  work  of  compression  and  of  forcing  the  gas  out  of 
the  compressor  will  be,  by  the  aid  of  equation  (368), 

U'  =  CIA D  =    fj,  vdp. 

4*[>'£j 

v  +  p'  v'  -  p"  v" 


A  «"n       » 

The  last  reduction  may  be  effected  by  finding  from  (368)  and  (366) 

p'  v'  —  p"  v"  =  (l  ("'  —  T")  —  b  (  ~rn  —  ~"~ 

If  the  vapor  be  saturated  at  state/",  and/  B  be  the  adia- 
batic of  superheated  vapor,  the  value  of  the  workfJ?  A  I) 
will  be  found  by  substituting  v,,  t» pv  respectively,  for  v", 
r",  p",  giving 


•  If  the  vapor  be  superheated  at  state  C,  its  volume  would 
be  given  by  equation  (364),  p  and  r  being  given.  If  it  be 
saturated,  *y,  will  be  given  by  equations  (362)  and  (350) 
when  either  p  or  r  are  given,  this  -c  being  r2  in  (369),  (370) 
and  (371),  when  the  initial  state  of  the  adiabatic  of  the 
superheated  vapor  is  on  the  curve  of  saturation. 

Since    C  I  is  an  adiabatic,  equation  (369)  will  give  the 
volume  v'  at  state  /,  and  (371)  the  pressure,  for  the  temper- 


346  REFRIGERATION.  [175.] 

ature  T',  This  pressure  is  assumed  to  be  constant  during 
condensation  ;  during  the  first  part,  from  /  to  B,  the  con- 
densation is  produced  by  a  reduction  of  the  temperature  and 
volume,  until  the  temperature  is  that  of  saturation,  r,,  un- 
der the  pressure  />„  while  from  B  to  A  the  temperature  and 
pressure  are  both  constant,  and  the  reduction  of  volume  is 
effected  by  the  condensation  of  vapor  to  a  liquid.  Assuming 
that  the  specific  heat  of  the  vapor  is  constant  at  constant 
pressure,  then,  for  complete  liquefaction,  the  heat  abstracted 
in  the  operation  /  A  will  be 

A,  =  ^(r'-r,)  +  Aei.  (396) 

Assume  that  heat  is  abstracted  from  the  liquid  at  con.-r;mr 
volume  from  state  A,  reducing  the  pressure  from  p^  to  pn 
and  temperature  from  rt  to  rt ;  the  path  of  the  fluid  will  be 
A  D,  and  the  heat  so  abstracted  will  be  H  D  A  G.  If  this 
heat  be  abstracted  from  the  circulating 
fluid  in  the  cold  room,  then  will  the  room 
absorb  that  amount  of  heat,  and  in  the 
evaporation  and  expansion  afterward  an 
equal  amount  of  heat  must  be  supplied 
from  the  cold  room  at  the  lower  tem- 
perature before  any  useful  amount  can  be  absorbed  by  the 
circulating  fluid  ;  and  if  H  D  j  g  be  the  amount  so  absorbed 
we  have 

II D  A  G  =  II  Djg. 

The  heat  emitted  will  be 

HD  A  G  =  c(rt-  rf). 

Let  II D  j  g  be  the  latent  heat  of  evaporation  in  the  oj.th 
part  of  a  pound  of  vapor  at  the  temperature  r, ;  then 

*.  *«  =  c  (T,  -  rt).  (397) 

(This  value  of  x«  is  the  same  as  x4  in  (380),  when  u"  =  0  ;  but  exceeds 
*4  in  Eq.  (385).) 


[176.]  THE   EFFICIENCY.  347 

The  refrigeration  per  pound  per  revolution  will  be 
A,  =   g  j  C  K, 


=  AM  ~  c(r,-  T,)  +  c'p  (r"  -  r&  (398) 

=  *«  +  0  (r2  -  r.)  -  c  (r,  -  r.)  +  ep  (r"  -  T,), 

=  h  _  c>  (r,  -  r.)  +  cp  (r"    -  r,),  (399) 

in  which  r0  is  the  absolute  temperature  of  melting  ice,  and 
h  the  total  heat  of  vapor  above  r0,  Article  85,  Eq.  (93). 
Then  equations  (396)  and  (397), 

A,  -  A,  -  Aei  +  <?„  (r'  -  7,)  -  Ae2  -  <>p  (r-  -  r2)  +  C(r,  -  r2) 
A  IK  -  A.,  -  £P  (r"  -  rs) 


h,\  (400) 

which  is  the  equivalent  of  equation  (391)  or  of  (395). 
176.  The  efficiency  will  be 


EXERCISE. 

If  the  inferior  absolute  pressure  of  the  ammonia  be  29' 
pounds  per  square  inch,  temperature  when  it  leaves  the 
cooling  coils  and  enters  the  compressor,  36°  F.,  then  com- 
pressed adiabatically  to  a  temperature  of  117°  F.,  then  con- 
densed to  saturation  and  to  a  liquid  at  the  constant  pressure 
corresponding  to  the  117°  F.,  then  admitted  to  the  cooling 
coils  and  the  temperature  reduced  to  that  corresponding  to 
the  initial  absolute  pressure,  29  pounds,  then  evaporated  and 
heated  to  the  initial  state  ;  find  the  thermal  units  of  refriger- 
ation and  of  condensation,  the  specific  heat  of  the  liquid 
being  1.08  ;  also  the  efficiency. 


348  REFRIGERATION. 

Retain  results  only  to  the  nearest  tenth. 

Inferior  temperature,  Eq.  (350) ra  =  458.7. 

Hence,  T*  =  —  1.9. 

Absolute  temperature  at  state  C.  460.6  +  36 r"  =  496.6. 

Superheating r"  —  r2  =    37.9. 

Greatest  volume  of  a  pound,  Eq.  (357)  for  p  =  4176,  v"  =      9.8. 

At  /,  absolute  temperature,  117  +  460.6 r'  =  577.6. 

"      volume,  Eq.  (369)  v'  =      2.4. 

"     pressure,  Eq.  (370) p'  =     118. 

From  B  to  A,  absolute  temperature,  Eq.  (350). ...  r,  =  527.8. 

Hence,  T,  =    67.2. 

Fall  of  temperature  from  A  to  D Ti  —  T*  —    69.1. 

Heat  absorbed  during  this  fall  of  temperature 1.08  X  69.1  =     74.6. 

Latent  heat  of  evaporation  at  —  1°. 9  P..  Eq.  (360).  flrt  =      557. 

"       "     "            "          "      66.2,        "       "  h,t  =     515. 

Heat  rejected  during  condensation,  Eq.  (396) hi  =  540.8. 

Refrigeration  per  lb.,  Eq.  (398) Jit  =  500.7. 

Work  done  by  the  compressor,  per  lb.,  Eq.  (400).  U  =  31198. 

Efficiency,  Eq.  (401) E,  =     12.5. 

177.  Experimental  Results.  The  following  data 
and  results  are  taken  from  the  report  of  a  test  of  a  I)e  La 
Yergne  refrigerating  plant  by  Messrs.  R.  M.  Anderson  and 
C.  H.  Page,  Jr.,  having  a  nominal  ice-melting  capacity  of 
about  110  tons  in  24  hours.  It  was  in  its  every  -day  work- 
ing condition  and  was  run  at  about  two-thirds  its  ordinary 
capacity.  Only  the  ice  plant  was  involved  in  the  experiment. 
It  consisted  of  two  single-acting  vertical  compressor  cylinders, 
driven  by  a  horizontal  double-acting  Corliss  engine,  as  shown 
in  Fig.  80,  a  feed  pump  and  a  condenser.  The  test  was 
during  11  hours  and  30  minutes.  Two  tests  were  made 
of  the  boilers,  as  the  first  indicated  such  high  efficiency 
it  was  considered  advisable  to  check  the  results.  It  will 
be  seen  that  the  second  test  also  gave  a  high  efficiency. 
The  second  test  was  during  12  hours.  All  the  instruments 
used  in  the  test  were  carefully  standardized.  (Thesis  1887.) 

FUEL,  FURNACE  AND  BOILERS. 

There  were  two  double  return  flue  boilers  arranged  to  run,  automati- 
cally, between  60  Ibs.  and  70  Ibs.  pressure  (gauge). 


[176.]                         EXPERIMENTAL   RESULTS.  349 

Fuel.  Lebigh  nut  (anthracite),  heat  of  combustion,  B.  T.  U.. . .  12229.6. 

Coal  for  12  hours,  Ibs  4422. 

Wood  for  starting  377  Ibs.  0.4  coal  equivalent,  Ibs 150.8. 

Coal,  total  equivalent,  Ibs 4572.8. 

Unburnt  coal,  Ibs 125.0. 

Coal  consumed,  Ibs 4447.8. 

Combustible,  total,  Ibs 3577.8. 

Coal  burned  per  hour,  Ibs 360.65. 

Heat  in  360.65  Ibs.  of  coal,  360.65  X  12229.6  B.  T.  U 4532901. 

Coal  per  H.  P.  per  h.  (2d  test,  H.  P.  was  102.92) 3.6013. 

Furnace,  grate  area,  sq.  ft 39. 

Ratio  of  heating  surface  to  grate  surface 35.63. 

Boilers,  heating  surface,  sq.  ft 1389. 

Water  fed  per  hour,  Ibs 3559.7.. 

Evaporation  per  Ib.  coal  fired,  Ibs 9.341. 

"     "       "       "      from  and  at  212° 9.957. 

"     "       "     consumed 9.601. 

"     "       "            "      from  and  at  212° 10.226. 

"  "  combustible,  from  and  at  212° 12.703. 

Average  gauge  pressure,  Ibs 66.0. 

Temperature  of  fire-room,  deg.  F 81. 

Average  temperature  of  flue  boiler,  deg.  F 319.28. 

"feed-water  "  "... 180.2. 

Total  heat  in  3559.7  Ibs.  steam  above  180°.2  F.,  B.  T.  U. . .  3661463. 

Efficiency  of  furnace  and  boiler,  per  cent,  8<y^300  =  80.7. 

ENGINE  (CORLISS). 

Piston,  diameter  of,  inches 32. 

"        stroke       "      "       36. 

"  speed  per  minute,  mean,  ft 194.988. 

Revolutions  per  minute,  average 31. 720. 

Average  indicated  horse-power 91.13. 

Water  consumed  per  IHF.  per  hour,  Ibs 34. 104. 

Steam  per  IHP.  per  hour,  Ibs 25.79. 

Steam  condensed  in  the  engine,  per  cent 24.35. 

Average  ratio  of  expansion 5.807. 

Steam  consumption  by  the  feed-pump  per  hour,  Ibs 50. 25. 

"  "  engine  per  hour,  Ibs 3509.45. 

Total  heat  in  3509.45  Ibs.  steam  above  180°. 2  F.,  B.  T.  U. . .  -  3609479. 
Heat  changed  to  work  per  h.,  B.  T.  U.,  102.92  X  1980000  •*•  778  =  261930. 

Efficiency  of  fluid 8609479  =  0'07ai- 


350                                      REFRIGERATION.  [176.] 

Coal  per  I.H.P.  per  hour  for  engine  and  pump 3.60. 

Combustible  for  I.H.P.  per  hour,  Ibs. 2.91. 

COMPRESSOR. 

Number  of  cylinders,  single-acting 2. 

Length  of  stroke,  inches.         36. 

Diameter  of  pistons,  each,  inches 18. 

Area  head  end  of  pistons,  each,  sq.  in 254.47. 

Average  number  of  revolutions  per  m 31.720. 

Piston  displacement  per  stroke,  cu.  ft 5.301. 

"    hour,  both,  cu.  ft 20179. 

Volume  of  sealing  oil  per  hour,  cu.  ft 143.8. 

Volume  filled  with  gas  per  hour,  both,  cu.  ft 20036. 

Indicated  horse-power,  mean 76.0892. 

Heat  eq.  of  work  by  compressor  per  hour,  B.  T.  U 193645. 

Efficiency  of  compressor  from  coal =  0.0407. 

"         "   mechanism 0.755. 

Temperature  of  ammonia  entering  compressor,  Deg.  F 57. 7. 

leaving                         "       116.1. 

Absolute  pressure  entering  compressor,  Ibs 28.88. 

leaving           "            " 132.01. 


CfornyoressoT*' 

J3 


35.014Q 


FIG.   88. 

Fig.  88  is  an  exact  copy  of  one  of  the  indicator  cards  taken  from  one 
of  the  compressor  cylinders.  The  lines  nearly  vertical  at  the  upper  part 
of  the  diagram  are  due  to  the  oscillations  of  the  indicator  spring. 


[177.]  EXPERIMENTAL   RESULTS.  351 

REFRIGERATION. 

Pressure  in  cooling  coil,  Ibs.  absolute 28.88. 

Hence,  temp,  of  liquid,  Deg.  F —  2.0. 

Temp,  compressed  gas,  Deg.  F 116.1. 

"       of  gas  entering  the  compressor,  Deg.  F 57.7. 

Rise  of  temperature  due  to  compression,  Deg.  F 58.4. 

Latent  heat  of  evaporation  of  1  Ib.  at  —  2.0  F.  Eq.  (360) 556.7. 

Superheating  in  cooling  coils,  37.9°  F. ,  B.  T.  U 19.27. 

Fall  of  temp,  of  liquid  in  cold  room,  Deg.  F. 69.4. 

Heat  imparted  to  cold  room  by  this  fall,  1.08  X  69.1 74.95. 

Heat  removed  from  cold  room  per  Ib.,  556.7  +  19.27  -  74.95  =  501.02. 

Ammonia  evaporated  per  hour,  Ibs 1669.5. 

"      "      B.  T.  U 851311. 

"  Ice-melting  capacity"  per  hour,  Ibs 5995. 

for  24  hours,  tons  (each  2000  Ibs.). . .  71 .95. 

per  IHP.  per  hour,  Ibs 65.79. 

"  Ib.  of  coal  (3.6013  per  HP.),  Ibs.. .  18.26. 

"  10  Ibs.  steam  (182.6  -»-  9.601)  Ibs. .  19.02. 

"  Ib.  combustible  (65.79  •*•  2.91),  Ibs.  22.61. 

"  Ib.  ammonia  evap.,  Ibs 3.59. 

EFFICIENCIES. 

Efficiency  of  furnace  and  boiler  (see  above) 0.807. 

"         "  steam  utilized  by  engine  (see  above) 0.0721. 

"         "  engine  referred  to  coal 0.0582. 

"         "  compressor  referred  to  engine  (see  above). 0.755. 

"          "            "               "        "  coal  (775  X  0.0582) 0.0439. 

QPjl  O-J  -J 

"         "  refrigeration  referred  to  compressor 193645  =      4>39- 

"  IHP.  of  engine 3.31. 

"  boiler.... 3.67  X  0.0643=     0.238. 

"coal 0.236X0.807=  0.192. 

that  is,  for  every  thermal  unit  in  the  coal  there  was  abstracted  0.19  of  a 
thermal  unit  from  the  cold  room. 

In  actual  ice-making,  only  about  30  to  45  per  cent  of  the 
pounds  of  ice-melting  capacity  can  be  produced  as  hard  ice 
suitable  for  commercial  purposes,  giving,  in  this  case,  be- 
tween 5.0  and  7.5  pounds  of  hard  ice. 

The  experiments  of  Professor  M.  Schroter  gave  from  19.1 
pounds  to  37. 4  pounds  of  ice  (net)  per  hour  per  indicated 


'352  REFRIGERATION.  [178.] 

horse-power  of  the  engine.  (One  result  is  given  as  48.8 
pound?,  but  the  test  was  of  too  short  duration  to  be  reliable.) 

If  4  pounds  of  coal  were  used  per  H.  P.  per  hour,  then 
there  would  be  produced  about  4.8  to  9.3  pounds,  net,  per 
pound  of  coal  consumed. 

Ledoux  remarks  that  manufacturers  estimate  about  50 
pounds  per  horse-power  per  hour  measured  on  the  driving- 
shaft  ;  hence,  if  the  delivered  power  be  0.80  of  the  indi- 
cated, this  would  be  equivalent  to  about  45  pounds  of  ice 
per  indicated  horse-power.  M.  Schroter's  tests,  and  the 
following,  by  Mr.  Shreve,  show  that  this  is  too  high,  if 
commercial  ice  is  intended. 

The  amount  of  ice  made  depends  upon  many  conditions : 
as,  clearances  in  the  cylinders,  friction  of  mechanism,  speed 
of  engine,  losses  along  the  pipes,  losses  in  opening  the  valves,, 
leakage,  losses  by  unavoidable  radiation,  losses  at  cans  by 
water  unfrozen  and  ice  cleavages ;  and,  these  being  con 
sidered,  it  seems  advisable,  in  designing,  to  assume  less  than 
one-third  the  pounds  of  ice-melting  capacity  for  the  probable 
pounds  of  commercial  ice  to  be  produced. 

178.  Test  of  an  ice-making  plant.  An  ice- 
making  plant  of  the  Cincinnati  Ice  Manufacturing  &  Cold 
Storage  Co.  was  tested  in  1888,  by  Messrs.  A.  L.  Shreve 
and  L.  W.  Anderson,  chiefly  to  determine  the  amount  of 
solid  ice  which  could  be  manufactured  in  24  hours  with  the 
plant  running  under  normal  conditions.  The  plant  con- 
sisted of  two  25-ton  (nominal)  and  one  50-ton  ice-machines, 
boilers,  pumps,  etc.,  used  in  actual  ice-making.  While  the 
machinery  was  doing  its  regular  work,  at  a  certain  hour, 
the  steam  pressure  was  observed  to  be  75  pounds  (gauge), 
and  all  the  conditions  of  the  furnace,  engines,  and  plant 
generally  were  observed,  and  the  conditions  continued  as 
nearly  uniform  as  possible  for  24  hours,  during  which  time 
108.87  tons  (of  2000  Ibs.)  of  ice  were  drawn,  from  which 


[179.]  THE   ABSORPTION    SYSTEM.  353 

should  be  deducted  nearly  a  ton  on  account  of  the  terminal 
condition  being  slightly  different  from  the  initial.  The 
actual  ice  weighed  out  at  75  pounds'  steam  pressure,  then, 
may  be  taken  as  216,000  pounds. 

The  engines  were  not  tested  at  the  same  time,  so  that  the 
ice  per  horse-power  was  not  definitely  known  ;  but  an  inde- 
pendent test  of  the  boilers  and  engines  was  made  prior  to 
the  "  capacity  "  test  given  above,  from  which  it  appears  that 
at  75  pounds'  pressure,  the  three  engines  developed  about 
415  indicated  horse-power  ;  according  to  which  21  pounds  of 
solid  ice  were  weighed  per  indicated  horse-power  per  hour. 
The  actual  amount  may  have  been  several  per  cent  more  or 
less.  The  machines  were  new  and  bearings  large,  and  the 
engine  and  compressor  absorbed  an  average  of  about  52  per 
cent  of  the  indicated  horse-power  of  the  engine. 

ABSORPTION  SYSTEM. 

179.  The  absorption  system  depends  upon  the 
fact  that  water  will  absorb  many  times  its  volume  -of  am- 
monia gas  ;  at  59°  F.  it  will  absorb  727  times  its  volume. 
This  is  a  chemical  action,  and  therefore  generates  heat.  Ac- 
cording to  Favre  and  Silberman,  925.7  B.  T.  II.  will  be 
developed  for  each  pound  of  gas  absorbed.  This  action  is 
substituted  for  the  compressor  in  raising  the  temperature  of 
the  ammonia  after  leaving  the  cold  room.  According  to 
Oarius,  the  coefficient  of  solubility  of  ammonia  gas  in  water 
is  represented  by  the  empirical  formula  (t  being  Deg.  C.) 

ft  =  1049.62  -  29.4963  t  +  0.676873  f  -  0.0095621  t* ; 
according  to  which  the  solubility  diminishes  as  the  temper- 
ature increases  and  soon  reaches  a  condition  at  which  it 
ceases  to  act ;  and  to  insure  continuous  working  the  absorp- 
tion chamber  is  cooled  by  water  externally. 

The  process  is  illustrated  in  Fig.  89.  A  solution  of  aqua- 
ammonia  being  in  the  generator  A  and  heated  by  means  of 
steam  passing  through  coils  in  this  chamber,  the  vapor  of 


354 


REFRIGERATION 


[179.] 


ammonia  is  generated  and  rises,  passing  through  tortnous 
ways  in  the  analyzer  C ;  thence  to  the  condenser  D.  The 
steam  which  rises  in  the 'analyzer  C  is  partly  condensed 
as  it  approaches  the  upper  end  of  the  vessel  and  falls  back 
to  the  generator,  and  that  which  passes  into  the  coils  over 
D  is  led  back  by  the  ammonia  drip,  so  that  nearly  pure 
ammonia  gas  enters  the  condenser  T).  Here  the  ammonia 
is  at  its  highest  pressure  and  temperature,  and  its  state  may 
be  represented  by  /?,  Fig.  00,  or  by  the  upper  right-hand 
corner  of  the  indicator  diagram  of  Fig.  88.  The  ammonia 
B  gas  passes  through  coils  of  pipes 

~\  in  the   condenser,  about   which 

|  \c  circulates   water ;   the   ammonia 

being  condensed  to  a  liquid  under 
a  constant  pressure,  the  path  of 
the  fluid  being  represented  by 
B  A,  Fig.  90.  The  liquid  passes 
to  the  lower  part  of  the  coils, 
or  to  a  receiver  especially  provided,  and  thence  through 
a  cock,  by  which  the  reduction  of  pressure  is  regulated  as 


{180.]  TEST   OF   ABSORPTION   PLANT.  355 

it  passes  into  the  cooler  /?.  The  reduction  of  pressure  is 
represented  by  A  D,  and  of  evaporation  during  the  expan- 
sion by  D  J.  The  liquid  and  a  small  amount  of  saturated 
vapor  is  now  in  the  cooler  H,  where  the  liquid  vaporizes, 
absorbing  heat,  the  operation  being  represented  by  J  (7, 
Fig.  90,  or  the  corresponding  line  in  Fig.  88.  The  brine, 
which  circulates  in  the  cold  room,  is  cooled  by  the  cold  am 
inonia  as  it  passes  through  the  cooler. 

During  the  process  of  cooling  the  brine,  the  liquid  am- 
monia becomes  a  gas.  The  gas  passes  from  the  cooler  II  to 
the  absorber  K,  the  pressure  in  K being  kept  a  little  lower 
than  in  H  by  the  ammonia  pump,  shown  at  the  right,  which 
draws  its  supply  from  K  and  forces  it  into 'the  generator  A. 
Into  the  absorber  A",  water  is  forced  and  absorbs  a  large 
volume  of  the  gas,  as  stated  above,  generating  heat,  the  oper- 
ation for  which  being  represented  by  C  B,  Fig.  90,  or  the 
compression  line  in  Fig.  88.  In  this  system  pure  water  is 
not  used  in  the  absorber,  but  instead  thereof,  water  is  drawn 
from  the  lower  part  of  the  generator  A  by  the  pipe  Z,  con- 
t  lining  but  little  ammonia,  the  mixture  being  called  "weak 
ammoniacal  liquor/'  By  the  absorption  of  the  gas,  strong 
aqua-ammonia  is  formed,  which  is  pumped  back  into  the 
generator,  and  the  operation  repeated. 

It  will  be  seen  that  the  operation  completes  a  cycle,  and 
that  the  changes  in  the  states  of  the  ammonia  are  similar  to 
those  in  which  a  compressor  is  used ;  hence,  if  there  were 
no  losses  of  heat,  except  those  described,  the  efficiency  would 
be  the  same. 

The  vessel  K  is  kept  sufficiently  cool  to  facilitate  the 
chemical  action  by  means  of  water  flowing  over  it. 

ISO.  Test  of  absorption  plant.  Professor  J. 
E.  Dentoii  made  a  seven  days'  continuous  test  of  an  absorp- 
tion plant  with  the  following  results.*  Every  element  en- 

*  Tram.  Am.  Soc.  Mech.  Eug.,Vol.  X .,  May,  1889. 


356  REFRIGERATION.  [180. } 

tering  into  the  problem  was,  as  far  as  practicable,  directly 
measured. 

Average  pressures,  above  atmosphere,  generator,  Ibs.  per.  sq.  in.     150.77. 
"  "  "  steam,         "     "        "  47.70. 


absorber,    "     "        "  23.4. 

Average  temperatures,  Deg.  F. ,  Generator 272. 

"                                    "       Condenser,  inlet 54J. 

outlet. 80. 

"                   "                "          range 25$. 

"      Brine,  inlet  21.20. 

"     outlet 16.14. 

range 506. 

"       Absorber,  inlet 80. 

"         outlet  111. 

range 31. 

Heater,  upper  outlet  to  generator  212. 

"     lower       "        absorber.  178. 

"     inlet  from  absorber 132. 

"       Inlet  from  generator 272. 

Water  returned  to  main  boilers..  260. 

Steam  per  hour  for  boiler  and  ammonia  pump,  Ibs 1986. 

Brine  circulated  per  hour,  cu.  ft 1633.7. 

"      "      pounds 119260. 

"    Specific  heat  0.800. 

"    Heat  eliminated  per  pound,  B.  T.  U 4.104. 

"    Cooling  capacity  per  24  hours,  tons  of  melting  ice 40.67. 

per  pound  of  steam,  B  T.  U 243. 

"    Ice-melting  capacity  per  10  Ibs.  of  steam,  Ibs. 17.1. 


Calorics,  refrigerating  effect  per  kilo,  of  steam  consumed 135. 

Heat  rejected  at  condenser  per  hour,  B.  T.  U  918000. 

"absorber     "        "        "        1116000. 

"    consumed  by  generator  per  Ib.  of  steam  condensed,  B.  T.  U.  932. 

Condensing  water  per  hour,  Ibs 36000. 

Condensing  coil,  approx.  sq.  ft.  of  surface 870. 

Absorber       •'           "        "     "     "        "     350. 

Steam             "          "        "     "    "        "     200. 

Pump,  Ammonia,  dia.  steam  cyl. ,  in 9. 

"             "            "    ammonia  cyl.,  in 3fc. 


[180.]  TEST   OF   ABSORPTION    PLANT.  357 

Pump,  Ammonia,  stroke,  in ..  10. 

"          revolutions  per  minute 22. 

"      Brine,  steam  cyl.,  diam.,  in i£. 

brine  cyl.,       "     b'. 

"      stroke,  in  10 

revolutions  per  m 70. 

Effective  stroke  of  puuips  0.8  of  full  stroke. 

18Oa.  Sulphur  Dioxide,  (or  Sulphurous  Acid). 
The  following  relations  have  been  found  for  this  acid : 
Specific  heat  of  the  gas,  0.15483. 

"      "    "    liquid,  0.4. 

Eelation   between  the  pressure  and  temperature  of  the 
saturated  vapor, 

1439.0       235629 
log  p  =  5.2330 5 (a) 

Equation  of  the  gas,  or  superheated  vapor, 

pv  =.  23.87  r  —  2  5J;9  °-  (b) 

Latent  heat  of  vaporization, 

Ae  =  171.26  —  0.25605  T  —  0.0013795  T\  (c) 

Volume  of  a  pound  of  the  saturated  vapor, 

778 
~~  2.3026 


Volume  of  a  pound  of  the  liquid, 
0.016 


1.484  —  0.0015659  T 

(Trans.  Soc.  Mech.  Eng.,  1890.) 


CHAPTER    VI. 

COMBUSTION 

181.  Essential  principle.  Combustion,  chemi- 
cally speaking,  is  the  combination  of  chemical  elements 
producing  heat.  Burning,  popularly  speaking,  is  the  ro 
suit  of  a  rapid  combination  of  oxygen  with  other  ele- 
ments. Carbon  and  hydrogen  are  the  chief  elements  of  the 
fuel  used  for  engineering  purposes.  Sulphur,  another  ele- 
ment, is  frequently  present,  but  is  comparatively  of  little 
value. 

When  two  substances  unite  chemically,  forming  a  sub- 
stance different  from  either,  it  is  said  that  a.  chemical  ofni'tij 
exists  between  them.  The  difference  between  n  mechanical 
mixture  and  a  chemical  combination  may  be  illustrated  in 
the  case  of  gunpowder.  The  process  of  manufacture  makes 
a  mechanical  mixture  of  charcoal,  sulphur,  and  nitre,  but  if 
the  gunpowder  be  fired  a  chemical  combination  results  and 
a  large  volume  of  gas  is  produced,  generating  a  large 
amount  of  heat  and  developing  a  strong  elastic  force  ;  and 
the  original  substance  entirely  disappears  and  new  sub- 
stances composed  of  different  combinations  of  the  original 
elements  are  formed. 

Definite  proportions.  In  every  chemical  compound  a 
definite  and  unvarying  proportion  of  its  elements  exists 
among  themselves. 

For  instance,  in  water  there  is  always  8  times  as  much 
oxygen  by  weight  as  there  is  of  hydrogen,  so  that  in  l"i> 


[181.] 


ESSENTIAL   PRINCIPLE. 


359 


pounds  of  water  there  is  88.8  pounds  of  oxygen  and  11.1 
pounds  of  hydrogen.  Any  chemical  compound  of  oxy- 
gen and  hydrogen  in  other  proportions  would  be  a  sub- 
stance entirely  different  from  water  ;  or,  if  in  the  100  parts 
there  were  some  other  element,  as  carbon,  the  substance 
would  also  be  different  from  water. 

The  chemical  equivalent  or  atomic  weight  is  expressed  by 
a  definite  number,  and  the  chemical  principle  of  definite 
proportions  may  be  expressed  in  the  form  of  the  two  follow- 
ing laws  : 

1.  The  proportions  by  weight  in  which  substances  com- 
bine  chemically   can  all  be   expressed   by   their  chemical 
equivalents,  or  by  simple  multiples  of  their  chemical  equiva- 
lents. 

2.  The  chemical  equivalent  of  a  compound  is  the  sum  of 
the  chemical  equivalents  of  its  constituents. 

Perfect  gases  at  a  given  pressure  and  temperature  com- 
bine in  proportion  to  their  volume. 

Neglecting  fractions  the  following  are  the  chemical 
equivalents  for  the  principal  elementary  constituents  with 
which  we  have  to  deal  in  fuel  and  air  : 


TABLE   I. 


Chemical  e 

juivalent. 

By  weight. 

By  volume. 

o 

16 

1 

N 

14 

1 

H 

1 

1 

c 

12 

? 

s 

32 

The  composition  of  a  compound  substance  is  indicated  by 
writing  the  symbol  of  the  elements  one  after  the  other,  and 


360 


COMBUSTION. 


[182.] 


affixing  to  each  symbol,  in  the  form  of  a  subscript  tue  num- 
ber of  its  equivalents  which  enter  into  one  equivalent  of  the 
compound.  Thus,  water  contains  two  chemical  equivalents 
of  hydrogen  to  one  of  oxygen, -and  is  indicated  by  the  ex- 
pression Ha  O  ;  and  the  constituents  by  weight  will  be  2  II 
-f-  16  O.  Similarly,  C  Os,  carbonic  acid,  contains  one  equiva- 
lent of  carbon  and  two  of  oxygen,  and  by  weight  12  C  -f- 
32  O. 

The  following  table  gives  the  composition  of  several  sub- 
stances : 

TABLE     II. 


Name. 

iU 

Proportions  of 
elenientH  by 
weight. 

ft 

1 

I 

Air  

H,0 
NH, 
CO 
C  O-, 
C,H4 
CH4 

so» 

SH, 
8,C 

N77H 
H2- 
H3- 
C  12- 
C  12- 
C  12- 
C  12- 
S  32- 
S  32- 
S  64- 

-O23 
-  O  16 
-X  14 
-  O  16 
-  O32 
-H2 
-114 
>-  O32 

-C  12 

100 
18 
17 
28 
44 
14 
16 
64 
84 
76 

N  79  O  21 
H  2  -(-  0 
H3  +  N 
C  +  0 
C  +  O2 
C  +  H2 
C+H4 

100 
2 
2 
2 
2 
2 
2 
2 
2 
2 

Water  

Ammonia  

Carbonic  oxide.  .  .  .... 
Carbonic  acid  

Oletiant  gas  
Marsh  gas  or  fire-damp. 
Sulphurous  acid  
Sulphuretted  hydrogen. 
Bisulphide  of  carbon. 

Air  is  not  a  chemical  compound,  but  a  mechanical  mixture  of  nitrogen 
and  oxygen. 

182.  The  heat  of  combustion  of  one  pound  of  a 
substance  combining  with  sufficient  oxygen  to  completely 
burn  it  has  been  found  by  experiment.  The  usual  process 
is  to  surround  a  small  furnace  with  a  quantity  of  water  so 
arranged  as  to  prevent  the  escape  of  heat ;  the  increased 


[182.] 


THE   HEAT   OF   COMBUSTION. 


361 


temperature  produced  in  the  water  by  the  burning  of  each 
pound  of  the  fuel  in  the  furnace  being  a  measure  of  the 
"  heat  of  combustion."  The  results  of  the  experiments  of 
Favre  and  Silberman  are  given  in  the  following  table  : 


TABLE  III 

SHOWING  THE  TOTAL  QUANTITIES  OF  HEAT  EVOLVED  BY  THE  COM- 
PLETE COMBUSTION  OF  ONE  POUND  OF  COMBUSTIBLE  WITH  OXYGEN  ; 
•  ADAPTED  FROM  THE  RESULTS  OBTAINED  BY  FAVKE  AND  SlLBETt- 
MAN.  THE  UNIT  OF  WEIGHT  IN  THIS  TABLE  BEING  ONE  POUND, 
AND  THE  UNIT  OF  TEMPERATURE  ONE  DEGREE  FAHR.  (FROM  39° 
TO  40°). 


Substance. 

Formula. 

Product. 

Unite  of 
heat. 

GAbES. 

H 

HaO 

62  032 

Carbonic  oxide  
Marsh  gas             

CO 
C  H4 

COa 
C  O2  &  Hs  O 

4,325 
23  513 

Oleflant  "-as     

C2  H4 

C  Oa  &  Ha  O 

21  343 

LIQUIDS. 

Oil  of  turpentine      

ClO    H,6 

COa&Ha  0 

19  533 

C2  H6  O 

C  Oa  &  Ha  O 

12  931 

Spermaceti  (solid)  

C,,  H64  Oo 

C  Oa  &  Ha  O 

18  616 

C  Sa 

0'  Oa  &  S  O' 

6  122 

SOLIDS. 

Carbon  (wood  charcoal)  

C 

(CO 

(C  O2 

4,451 
14,544 

14  485 

13  972 

Native  Graphite 

14  035 

s 

S  Oa 

4048 

Phosphorus  (observed  by  Andrews.).  . 

p 

Pa05 

10,715 

The  heat  units  in  a  pound  of  fuel  will  be  nearly  the  sum 
of  the  heat  units  of  combustion  of  its  constituents.  Take, 
for  example,  oleh'ant  gas  :  According  to  Table  II.  the 
chemical  equivalents  by  weight  are  14,  of  which  12  are  carbon 


362  COMBUSTION.  [182.] 

and  2,  hydrogen  ;  or,  j  of  the  compound  is  carbon  and  \ 
hydrogen.     Then,  from  Table  III.,  we  find 

\  Ib.  H  gives  }  of  62032    =     8862  B.  T.  U. 
f  Ib.  carbon  gives  f  of  14554  =  12475  "  .  "     "  ' 

Total  ........................  =  21337"    "     " 

which  is  only  6  thermal  units  less  than  the  value  given  in 
the  table,  as  deduced  from  experiment. 

If  the  principal  constituents  are  carbon,  hydrogen,  and 
oxygen,  it  is  found  that  the  total  heat  of  combustion  in 
B.  T.  U.  will  be  given  nearly  by  the  following  formula  : 

A  =  14500  (C  +  4.28  (//  -  i  0)  ), 


in  which  4.28  =  ,  reduces  the  hydrogen  to  an  equiva- 

14500 

lent  of  carbon,  and  £  0  is  deducted  from  the  hydrogen,  for 
it  is  assumed  that  the  oxygen  present  unites  with  the  hydro- 
gen, forming  water.  Such  substances  are  called  hydro- 
carbons. 

The  total  heat  of  combustion  is  usually  computed  from 
its  chemical  analysis,  as  shown  on  page  261. 

The  following  table  gives  the  total  heat  of  combustion  of 
certain  fuels  (see  Journal  of  United  Semice  Institution, 
Eng.,  Vol.  XL,  1867  ;  Box  On  Heat,  p.  60).  The  speci- 
mens were  of  the  best  quality,  and  are  too  high  for  ordinary 
practice.  Commercial  coal  of  similar  grade  would  be  about 
0.7  to  0.8  of  these  values.  Commercial  Lehigh  (anthracite), 
analyzed  at  the  Institute,  gave  12229  B.  T.  U. 


[183.  J 


THE   INCOMBUSTIBLE   MATTER. 


363 


TABLE    IY. 

TOTAL   HEAT   OF   COMBUSTION   OF   FUEL. 


FUEL. 

Carbon. 

1 

I 

Equivalent  to 
pure  carbon. 

II 

1 

E 
14 
12 
14 
13.2 
123 

15.75 
15.9 
15.4 
153 
14.25 
16.0 
15.15 
15.6 
13.65 
12.15 
10.0 
7.25 
7.5 
5.8 

22.7 
22.5 

Total  heat  of 
combustion. 

I.  CHARCOAL  —  from  wood  
•'           from  peat.  . 

c. 

0.93 

0.94 

0.88 
0.82 

0.915 
0.90 
0.87 
0.80 
0.77 
0.88 
0.81 
0.84 
0.77 
0.70 
0.58 

0.50 

0.84 
0.85 

H. 

0.035 
0.04 
0.04 
0.054 
0.05 
0.052 
0.052 
0.056 
0.052 
0.05 
006 

0.16 
0.15 

0. 

0.026 
0.02 
0.03 
0.016 
0.06 
0.054 
0,04 
0.08 
0.15 
0.20 
0.31 

0 
0 

C'. 
0.93 
0.80 
0.94 

0.88 
0.82 

105 
1.06 
1.025 
1  02 
0.95 
1.075 
1.01 
1.04 
0.91 
0.81 
0.66 

0.50 

1.52 
1.49 

h. 
13500 
11600 
13620 
12760 
11890 

15225 
15310 
14860 
14790 
18775 
15837 
14645 
15080 
13195 
11745 
9660 
7000 
7245 
5600 

21930 
21735 

II  COKE  —  good      

"        middling  .  . 

bad  ...     

III.  COAL- 
1    Anthracite     

2    Dry  bituminous  .  . 

3     "             •<         

4     "             "         

5     "             "         

6.  Caking  

8    Cannel  

10.  Lignite  

IV.  PEAT  —  dry  

"        containing  25$  moisture  .. 
V    WOOD  —  Dry   

"        containing  20$  moisture.  . 
VI.  MINERAL  OIL  — 
from  

to  

183.  The  incombustible  matter  is  called  ash. 
The  principal  ingredients  of  ash  are  shown  in  the  following 
analysis,  which  is  from  the  geological  survey  of  Ohio : 

Bituminous  coal.     Percentage  of  ash,  5.15. 

Silica 58.75 

Alumina 35.30 

Sesquioxide  of  iron 2.09 

Lime 1.20 


364 


COMBUSTION. 


[184.] 


Magnesia 0.68 

Potash  ami  sochv 1 .08 

Phosphoric  acid 0. 13 

Sulphuric  acid 0,24 

Sulphur,  combined 0.41 


The  proportions  vary  greatly  with  different  fuels. 


99.88 


184.  Air  required  for  combustion.  Consider 
pure  carbon.  The  chemical  equivalent  of  oxygen  is,  ac- 
cording to  Table  I.,  |f  of  that  of  carbon.  If  the  carbon  be 
completely  burned,  C  Ot  is  formed,  so  that  the  proportion 
by  weight  will  be  ff  of  oxygen  to  1  of  carbon.  According 
to  Table  II.,  0.23  of  the  air  by  weight  is  oxygen  ;  hence 

Weight  of  air  jter  U>.  of  carbon  =  ?$  -f-  0.23  =  12  Ibs .,  nearly. 

If  the  compound  contains  carbon,  hydrogen  and  oxygen, 
we  will  have,  nearly, 

Weight  of  „•,,•}.<>,>  //>.  fuel  =  A  =  12  C  +  30  (//— ti  O). 

The  following  table,  computed  from  this  formula,  is  given 
by  Kankine. 

TABLE    V. 


I.  CHARCOAL  —  from  wood  
'  '            from  peat  

0.93 
0  80 

11.16 
9  6 

II.  COKE  —  good  

0.94 

1  1  28 

III.  COAL  —  anthracite  .-  

0  915 

0  035 

0026 

12.13 

dry  bituminous  

087 

0.05 

0.04 

12  06 

caking  

0.85 

0  05 

0.06 

11.73 

0  75 

005 

0.05 

1058 

cannel  

0  84 

0  06 

0  08 

11.88 

dry  long  flaming  

0  77 

0  05 

0.15 

10.  ?2 

lignite  ,    

0  70 

005 

0.20 

9.30 

IV.  PEAT—  dry  

0  58 

006 

0.31 

7.68 

V.  WOOD—  dry  . 

0  50 

6  00 

VI.  MINER  \L  OIL 

0  85 

015 

o 

15  65 

[185,  186.]  TEMPERATURE   OF   FIRE.  365 

Besides  the  air  necessary  to  furnish  the  oxygen,  some  is 
required  for  dilution,  so  as  to  secure  a  more  free  access  of 
air  to  the  fuel.  The  above  table  shows  that  about  12  pounds 
of  air  is  required  per  pound  of  fuel,  and  experiment  indi- 
cates that  from  1£  to  2  times  this  amount  is  required  in  the 
furnace  for  combustion  and  dilution.  An  excess  of  air 
causes  a  waste  of  heat  by  transporting  an  unnecessary  amount 
of  heat  up  the  chimney,  and  a  deficiency  causes  imperfect  com- 
bustion. Forced  draft  requires  less  air  than  a  natural  draft. 

185.  Forced  draft.     Most  American  sea-going  steam- 
ers have  boilers  designed  to  burn  anthracite  coal  with  natural 
draft,  and  5  to  5|  pounds  of  coal  is  burned  per  hour  per 
square  foot  of  grate.     Torpedo-boats,  with  bituminous  coal 
and  forced  draft  of  6  inches  of  water,  may  burn  96  pounds 
per  square  foot  of  grate  per  hour. 

In  a  furnace  in  which  19  pounds  of  anthracite  coal  were 
burned,  a  forced  draft  by  means  of  a  screw  revolving  in  the 
chimney  was  introduced,  causing  a  burning  of  38^  pounds 
of  coal,  and  a  production  of  80  per  cent  more  steam  than  in 
the  former  case.  In  18-17  Robert  L.  Stevens  introduced 
the  plan  of  air-tight  fire-rooms,  by  which  means  a  forced 
draft  was  produced  by  forcing  air  into  the  room  occupied 
by  the  firemen. 

186.  Temperature  of  fire.     If  the  volume  be  con- 
stant, as  in  case  of  an  explosion  in  a  closed  vessel,  then  will 
the  rise  of  temperature  be  given  by  equation  (37),  page  55  ; 
but  if  the  pressure  be  constant,  as  in  the  ordinary  furnace, 
then  will  it  be  given  by  equation  (38). 

Take  the  case  of  pure  carbon  burned  with  2-1  pounds  of 
air.  The  total  heat  of  combustion  will  be,  Table  III.,  14544 
thermal  units.  This  heat  is  expended  in  heating  25  pounds 
of  matter,  of  which  24  pounds  is  air,  whose  specific  heat  is 
0.238,  and  one  pound  of  carbonic  acid  gas,  whose  specific 
heat  is  0.217.  Call  the  specific  heat  of  the  mixture  0.237  ; 
then 


366  COMBUSTION.  [187.] 


The  only  measurement  with  a  pyrometer  which  has  come 
to  my  notice  gives  a  much  lower  temperature  than  is  found 
by  this  formula. 

187.  Height  of  chimney.  The  height  of  the 
chimney  to  produce  a  natural  draft  must  be  such  that  the 
difference  between  the  weight  of  a  column  of  the  hot  gases, 
having  one  square  foot  for  its  base  and  height  equal  to  the 
height  of  the  chimney,  and  that  of  a  column  of  equal  height 
of  external  air,  shall  produce  the  required  velocity  of  air  in 
the  chimney. 

Let  w0  be  the  weight  of  fuel  burned  in  the  furnace  per 

second, 
F0,  the  volume  of  the  air  at  32°  supplied  per  pound 

of  fuel  burned, 

TO,  the  absolute  temperature  at  32°, 
A,  the  area  of  the  cross-section  of  the  chimney, 
m,  A  -r-  perimeter  of  chimney, 
T,,  the  absolute  temperature  of  the  gases  discharged 

from  the  chimney, 

w,  the  weight  of  a  cubic  foot  of  the  hot  gases, 
I,  the  length  from  the  furnace  to  the  top  of  chimney, 
u,  the  velocity  of  the  current  in  the  chimney  per 

second  ; 

then,  if  24  pounds  of  air  be  supplied  per  pound  of  fuel, 
F0  =  25  X  12  =  300  cu.  ft. 

u  A  —  wn  V,  ^  =  volume  of  gases  per  second  ; 


4*, 

w  =  Is  (0.0807 


[187. J  HEIGHT   OF   CHIMNEY.  367 

Accordiiig  to  Peclet,  the  head  to  produce  the  velocity  u 
for  20  Ibs.  coal  burned  per  sq.  ft.  of  grate  per  hour  will  be 


Having  A,  the  height  of  the  chimney  may  be  found. 

Let  H  be  the  height  of  chimney, 
rs,  the  absolute  temperature  of  the  external  air ;  then 

Weight  of  the  column  of  air      =  0.0807 .  Is  H. 


"  gases  =  f  0.0807  +  -y  j  -*-H. 
fi        "a  column  of  gases  of  length  h 

=  (0.0807 +^)1-*; 

.  • .  (o.0807  +  -L)  -°  (H  +  h)  =  0.0807  1«  H  • 

\  300/  Tj  Ty 


0.96  ~l  -1 


The  weight  discharged  per  second  will  be 


wu  =  ^  [0.0807  + 


=  constant 


|/0.96  L  - 


This  will  be  a  maximum  for 


that  is,  the  lest  chimney  draft  takes  place  when  the  absolute 
temperature  of  the  gas  in  the  chimney  is  to  that  of  the  ex- 
ternal air  as  25  to  12. 


368  COMBUSTION.  [187.; 

This  in  the  preceding  equation  gives 
h  =  //. 

In  the  solution  for  a  maximum,  I  is  considered  constant, 
an  assumption  which  affects  the  result  by  only  a  small 
amount. 

The  height  of  a  chimney  is  often  determined  by  sur- 
rounding circumstances,  and  sometimes  by  imagined  con- 
ditions of  future  use ;  and  in  such  cases  are  not  subject  to 
computation. 

In  ordinary  practice  chimneys  are  from  forty  to  one  hun- 
dred and  twenty  feet.  Above  one  hundred  feet  the  effect 
of  additional  height  is  comparatively  small. 

The  tallest  chimney  of  which  we  have  knowledge  is  441.6 
feet  high,  eleven  feet  and  a  half  in  diameter  at  the  base, 
and  ten  feet  at  the  top.  It  was  built  by  the  Mechernich 
Lead  Mining  Co.  (Van  Nostrand's  Eng.  Mag.,  1886,  page 
264).  For  dimensions  of  large  chimneys,  see  Van  Xostrand's 
Eng.  Mag.,  September,  1883,  page  216 ;  also  Trans.  Am. 
Soc.  Civ.  Engineers,  1885  ;  also  No.  1,  Science  Series,  by 
D.  Van  Nostrand. 


APPENDIX  I. 


(Extract  from  an  article  by  the  author  in  the  Philosophical  Magazine  for 
November,  1885.*) 

THE  LUMINIFEROUS  ETHER. 

Two  properties  of  the  luminiferous  ether  appear  to  be  known  and 
measurable  with  a  high  degree  of  accuracy.  One  is  its  ability  to  transmit 
light  at  the  rate  of  186300  miles  per  second,  f  and  the  other  its  ability  to 
transmit  from  the  sun  to  the  earth  a  definite  amount  of  heat  energy. 

In  regard  to  the  latter,  Herschel  found,  from  a  series  of  experiments, 
that  the  direct  heat  of  the  sun,  received  on  a  body  at  the  earth  capable  of 
absorbing  and  retaining  it,  is  competent  to  melt  an  inch  in  thickness  of 
ice  every  two  hours  and  thirteen  minutes.  This  is  equivalent  to  nearly 
71  foot-pounds  of  energy  per  second. 

In  1838  M.  Pouillet  found  that  the  heat  energy  transmitted  from  the 
sun  to  the  earth  would,  if  none  were  absorbed  by  our  atmosphere,  raise 
1.76  grammes  of  water  1°  C.  in  one  minute  on  each  square  centimeter  of 
the  earth  normally  exposed  to  the  rays  of  the  sun.  { 

This  is  equivalent  to  83.5  foot-pounds  of  energy  per  second,  and  is  the 
value  used  by  Sir  William  Thomson  in  determining  the  probable  density 
of  the  ether.  §  Later  determinations  of  the  value  of  the  solar  constant  by 
MM.  Sorret,  Crova,  and  Violle  have  made  it  as  high  as  2.2  to  2.5  calories. 
But  the  most  recent,  as  well  as  the  most  reliable,  determination  is  by  Pro- 
fessor S.  P.  Langley,  who  brought  to  his  service  the  most  refined  ap- 
paratus yet  used  for  this  purpose,  and  secured  his  data  under  favorable 
conditions  ;  from  which  the  value  is  found  to  be  2.8  ±  calories  |  with 
some  uncertainty  still  remaining  in  regard  to  the  first  figure  of  the  deci- 

*  Published  in  Van  Nostrand's  Engineering  Magazine,  January,  1886.    Also  Snenct 
Series,  No  85. 

f  Professor  Michelson  found  the  velocity  of  light  to  be  299740  kilometers  per  sec- 
ond in  air,  and  299828  kilometers  in  a  vacuum,  giving  an  index  of  refraction  of 
1  000265.  (Journal  of  Arts  and  Science,  1879,  Vol.  XVIii.,  p.  390.) 

*  Camples  Rendus,  1838,  Tom.  VII.,  pp.  24-26. 

§  Trans.  Roy.  Soc.  of  Edinburgh,  Vol.  XXI.,  Part  I. 

I  Am.  Journ.  of  Arts  and  Science,  March,  1883,  p.  195.    Also  Comntes  Eendus. 


370  APPENDIX   I. 

mal.  We  will  consider  it  as  exactly  2.8  in  this  analysis,  according  to 
which,  there  being  7000  grains  in  a  pound  and  15.432  grains  in  a  gramme, 
we  have  for  the  equivalent  energy 

2.8  X  15.432  9  772  X  144 
7000  X  5  X  (U551T60 
per  second  for  each  square  foot  of  surface  normally  exposed  to  the  sun's 
rays,  which  value  we  will  use.  Beyond  these  facts,  no  progress  can  be 
made  without  an  assumption.  Computations  have  been  made  of  the 
density,  and  also  of  the  elasticity,  of  the  ether  founded  on  the  most  arbi- 
trary, and  in  some  cases  the  most  extravagant,  hypotheses.  Thus,  Her- 
Bchel  estimated  the  stress  (elasticity)  to  exceed 

17  X  109  =  (17,000,000,000)  pounds  per  square  inch  ;  * 
and  this  high  authority  has  doubtless  caused  it  to  be  widely  accepted  as 
approximately  correct.  But  his  analysis  was  founded  upon  the  assump- 
tion that  the  density  of  the  ether  was  the  same  as  that  of  air  at  sea-level, 
which  is  not  only  arbitrary,  but  so  contrary  to  what  we  should  ex]xrt 
from  its  non-resisting  qualities  as  to  leave  his  conclusion  of  no  value. 
That  author  also  erred  in  assuming  that  the  tensions  of  gases  were  as  the 
wave-velocities  in  each,  instead  of  the  mean  square  of  the  velocity  of  the 
molecules  of  a  self-agitated  gas  ;  but  this  is  unimportant,  as  it  happens  to 
be  a  matter  of  quality  rather  than  of  quantity.  Herschel  adds.  "  Consid- 
ered according  to  any  hypothesis,  it  is  impossible  to  escape  the  conclusion 
that  the  ether  is  under  great  stress."  We  hope  to  show  that  this  con- 
clusion is  not  warranted  ;  that  a  great  stress  necessitates  a  great  density  ; 
but  that  both  may  be  exceedingly  small.  A  great  density  of  the  ether 
not  only  presents  great  physical  difficulties,  but,  as  we  hope  to  show,  is 
inconsistent  with  the  uniform  elasticity  and  density  of  the  ether  which 
it  is  believed  to  possess  ;  and  every  consideration  would  lead  one  to  ac- 
cept the  lowest  density  consistent  with  those  qualities  which  would  enable 
it  to  perform  functions  producing  known  results. 

In  a  work  on  the  Physics  of  Ether,  by  S.  Tolver  Preston,  it  is  esti 
mated  that  the  probable  inferior  limit  of  the  tension  of  the  ether  is  500 
tons  per  square  inch,  a  very  small  value  compared  with  that  of  Herschel's. 
But  the  hypothesis  upon  which  this  author  founded  his  analysis  was— 
The  tension  of  the  ether  exceeds  the  force  necessary  to  separate  the 
atoms  of  oxygen  and  hydrogen  in  a  molecule  of  water  ;  as  if  the  atoms 
were  forced  together  by  the  pressure  of  the  ether,  as  two  Magdeburg 
hemispheres  are  forced  together  by  the  external  air  when  there  is  a 
vacuum  between  them.  This  assumption  is  also  gratuitous,  and  is  re- 
jected for  want  of  a  rational  foundation. 

Young  remarks  :  "  The  luminiferous  ether  pervading  all  space  is  not 

*  Familiar  Lecture*,  p.  282. 


THE   LUMINIFEROUS   ETHER.  871 

only  highly  clastic,  but  absolutely  solid."  *  We  are  not  certain  in  what 
sense  this  author  considered  it  as  solid  ;  but  if  it  be  in  the  sense  that  the 
particles  retain  their  relative  positions,  and  do  not  perform  excursions  as 
they  do  in  liquids,  jt  is  a  mere  hypothesis,  which  may  or  may  not  have  a 
real  existence.  If  it  be  in  the  sense  that  the  particles  suffer  less  resistance 
to  a  transverse  than  to  a  longitudinal  movement,  there  are  some  grounds 
for  the  statement,  as  shown  in  circularly-polarized  light.  Bars  of  solids 
are  more  easily  twisted  than  elongated,  and,  generally,  the  shearing  re- 
sistance is  less  than  for  a  direct  stress.  It  certainly  cannot  be  claimed 
that  the  compressibility  of  the  ether  (in  case  we  could  capture  a  quantity 
of  it)  is  less  than  that  of  solids. 

Sir  William  Thomson  made  a  more  plausible  hypothesis,  by  assuming 
that  "  the  maximum  displacement  of  the  molecules  of  the  ether  in  the 
transmission  of  heat  energy  was  ^  of  a  wave  length  of  light,  the  average 
of  which  maybe  taken  as  ^^y  of  an  inch."  Hence  the  displacement 
was  assumed  to  be  ^5^5^05  of  an  inch  ;  by  means  of  which  he  found  the 
weight  of  a  cubic  foot  to  be  £  X  10  ~20  of  a  pound,  f  We  also  notice  that 
Hr.  Belli  estimated  the  density  of  the  ether  to  be  £  X  10~ 1S  of  a  pound  ;  \ 
but  M.  Herwitz,  assuming  this  value  to  be  too  small  and  Thomson's  as 
too  large,  arbitrarily  assumed  it  as  10  ~18  of  a  pound  per  cubic  foot  ;  but 
arbitrary  values  are  of  small  account  unless  checked  by  actual  results. 

We  propose  to  treat  the  ether  as  if  it  conformed  to  the  Kinetic  Theory 
of  Gases,  and  determine  its  several  properties  on  the  conditions  that  it 
shall  transmit  a  wave  with  the  velocity  of  186300  miles  per  second,  and 
also  transmit  133  foot-pounds  of  energy  per  second  per  square  foot.  This 
is  equivalent  to  considering  it  as  gaseous  in  its  nature,  and  at  once  com- 
pels us  to  consider  it  as  molecular  ;  and,  indeed,  it  is  difficult  to  conceive 
of  a  medium  transmitting  light  and  energy  without  being  molecular. 
The  Electromagnetic  Theory  of  Light  suggested  by  Maxwell,  as  well  as 
the  views  of  Newton,  Thomson,  Herschel,  Preston,  and  others,  are  all  in 
keeping  with  the  molecular  hypothesis.  If  the  properties  which  we  find 
by  this  analysis  are  not  those  of  the  ether,  we  shall  at  least  have  deter- 
mined the  properties  of  a  substance  which  might  be  substituted  for  the 
ether,  and  secure  the  two  results  already  named.  It  may  be  asked,  Can 
the  Kinetic  theory,  which  is  applicable  to  gases  in  which  waves  are  propa- 
gated by  a  to-and-fro  motion  of  the  particles,  be  applicable  to  a  medium 
in  which  the  particles  have  a  transverse  movement,  whether  rectilinear, 
circular,  elliptical  or  irregular  ?  In  favor  of  such  an  application,  it  may 
be  stated  that  the  general  formulae  of  analysis  by  which  wave  motion  in 
general,  and  refraction,  reflection  and  polarization  in  particular,  aie  dis- 
cussed, are  fundamentally  the  same  ;  and  in  the  establishment  of  the 

*  Young's  Works,  Vol.  I.,  p.  415.  t  Phil.  Mag.,  1855  [4]  IX.,  p.  39. 

t  Cf .  Fartschritte  der  Physik,  1859. 


372  APPENDIX   I. 

equations  the  only  hypothesis  in  regard  to  the  path  of  a  particle  is—  It 
will  move  along  the  path  of  least  resistance.  The  expression  F*  x  <  -^  (5 
is  generally  true  for  all  elastic  media,  regardless  of  the  path  of  the  indi- 
vidual molecules.  Indeed,  granting  the  molecular  constitution  of  the 
ether,  is  it  not  probable  that  the  Kinetic  theory  applies  more  rigidly  to  it 
than  to  the  most  perfect  of  the  known  gases  ?  * 

The  133  foot-pounds  of  energy  per  second  is  the  solar  heat  energy  in  a 
prism  whose  base  is  1  square  foot  and  altitude  186300  miles,  the  distance 
passed  over  by  a  ray  in  one  second  ;  hence  the  energy  in  1  cubic  foot 
will  be 

d) 


Where  results  are  given  in  tenth-units  of  high  order,  as  in  the  last  ex- 
pression, it  seems  an  unnecessary  refinement  to  retain  more  than  two  or 
three  figures  to  the  left  hand  of  the  ten*  ;  and  we  will  write  such  expres- 
sions as  if  the}'  were  the  exact  results  of  the  computations. 

If  V  be  the  velocity  of  a  wave  in  an  elastic  medium  whose  coefficient 
of  elasticity,  or  in  other  words,  its  tension,  he  and  density  6,  both  for  the 
same  unit,  we  have  the  well-known  relation 


And  for  gases  we  have 

•  «*. 

where  y  =  1.4  ;    and  the  differential  of  the  latter  substituted  in  the 
former  gives 


The  tension  of  a  gas  varies  directly  as  the  kinetic  energy  of  its  mole- 
cules per  unit  of  volume.  If  «J  be  the  mean  square  of  the  velocities  of 
the  molecules  of  a  self  -agitated  gas,  we  have 

e  <x.  J  T*,  or  0*  =  x  -j.  •  (3) 

where  a;  is  a  factor  to  be  determined.     Equations  (2)  and  (3)  give 

t>»  =  ?  F».  (4) 

Assuming,  with  Clausius,  that  the  heat  energy  of  a  molecule  due  to  the 
action  of  its  constituent  atoms,  whether  of  rotation  or  otherwise,  is  a 
multiple  of  its  energy  of  translation,  we  have  for  the  energy  in  a  unit  of 
volume  producing  heat, 

*  See  also  remarks  by  G.  J.  Stoney,  PMl.  Mag.,  1868  [4]  XXXVI.,  pp.  188,  188. 


THE   LUMINIFEKOUS   ETHER.  373 


where  y  is  a  factor  to  be  determined.  If  c  be  the  specific  heat  of  a  gas, 
w  its  weight  per  cubic  foot  at  the  place  where  g  =  32.2,  J.  Joule's  me- 
chanical equivalent,  r  its  absolute  temperature  ;  then  the  essential  energy 
of  a  cubic  foot  of  the  medium  will  becwrj;  and  observing  that  w  =  gd, 
we  have 

4  y  6  v1  =  c  g  6  T  J,  (5) 

which,  reduced  by  (4),  gives 

(6) 

the  second  member  of  which  is  constant  for  a  given  gas.  To  find  its 
value  we  have 

Hydrogen.         Air.  Oxygen. 

Specific  heat,*    .....         3.4093         0.2375  0.2175 

Velocity  of  sound,  feet  per  second,  at  )         ^            ^  ^ 

T  =  4»O.^   ,         .  .  .  .  .  ) 

and  g  =  32.2,  y  =  1.4,  J  =  772.  These,  substituted  in  the  second 
member  of  (6),  give 

a;  y  for  hydrogen,      .....        6.599 

"      air  .......  6.706 

"      oxygen,         .....        6.596 

3)19.901 


Mean, 6.63 

This  value,  which  is  nearly  constant  for  the  more  perfect  gases,  we  pro- 
pose  to  call  the  modulus  of  the  gas,  and  represent  it  by  /*  ;  and  for  the  pur- 
poses  of  this  paper  we  will  use 

P  =  6.6. 

This  relation  of  the  product  x  y  being  a  constant,  has,  so  far  as  we  are 
informed,  been  overlooked  by  physicists,  and  is  worthy  of  special  notice, 
since  it  determines  the  value  of  one  of  the  factors  when  the  other  has 
been  found.  Kronig,  Clausius,f  and  Maxwell  give  for  x  the  constant 
number  3,  but  variable  values  for  y.  \ 

We  are  confident  that  the  value  of  x  is  not  strictly  constant ;  or  if  it 
is,  it  exceeds  3,  since  the  effect  of  the  viscosity  of  a  gas  would  necessitate 
a  larger  velocity  to  produce  a  given  tension  than  if  it  were  perfectly  free 

*  Stewart  on  Heat,  p.  «29. 
t  Phtt.  Mag.,  1857  [4]  XIV.,  p.  123. 

J  Theory  of  Heat,  pp.  314  and  317.  Maxwell  states  that  the  value  for  y  Is  probably 
equal  to  1.634  for  air  and  several  of  the  perfect  gases.  This  would  make  x  =  4  nearly. 


374  APPENDIX   I. 

from  internal  friction.  For  our  purpose,  it  will  be  unnecessary  to  find 
the  separate  values  of  x  and  y  ;  but  if  we  have  occasion  to  use  the  former 
in  making  general  illustrations,  we  will  call  it  3,  as  others  have  done 
heretofore.  If  the  correct  value  of  x  exceeds  3,  it  will  follow  that  the 
velocity  of  the  molecules  exceeds  the  values  heretofore  computed.  *  Ac- 
cording to  Thomson,  Stokes  showed  that  in  the  case  of  circularly  polar- 
ized light  the  energy  was  half  potential  and  half  kinetic  ;f  in  which  case 
y  =  2,  and  therefore  x  =  3.3. 

The  energy  in  a  cubic  foot  of  the  ether  at  the  earth  being  given  by 
(1)  and  (5),  we  have,  by  the  aid  of  (4), 


.     .  _  _  4  X  1.4  X  2  __    _  2__  (9) 

~  3  X  1C7  X  6.6  X  (186300  X  5280)*        35  X  10*4 

•which  is  the  mass  of  a  cubic  foot  of  the  ether  at  the  earth,  and  which 
would  weigh  at  the  place  where  g  =  32.2  about 

tc  =  Aj  of  a  pound,  (10) 

compared  with  which  Thomson's  value  is  less  than  4000  times  this  value. 
Thomson  remarked  that  the  density  could  hardly  be  100,000  times  as 
small  —  a  limit  so  generous  as  to  include  far  within  it  the  value  given  in 
(9).  According  to  equation  (10),  a  quantity  of  the  ether  whose  volume 
equals  that  of  the  earth,  would  weigh  about  ^  of  a  pound.  If  a  particle 
describes  the  circumference  of  a  circle  in  the  same  time  that  a  ray  passes 
over  a  wave-length  ?-,  the  radius  of  the  circle  will  be,  using  equation  (4), 


or  the  displacement  from  its  normal  position  will  be  about  ^  of  a  wave- 
length, or  about  mr'uoff  °f  an  *ncn  at  the  earth. 
Eliminating  V  between  (2)  and  (8)  gives 

(11) 

for  the  tension  of  the  ether  per  square  foot  at  the  earth,  and  is  equiva- 
lent to  about  1.1  of  a  pound  on  a  square  mile.  The  tension  of  the  atmos- 
phere at  sea-level  is  more  than  30,000,000,000  times  this  value.  It  some- 

*  Maxwell  gives  for  the  mean  square  of  the  velocities,  or,  in  other  words,  the  velocity 
whose  square  is  the  mean  of  the  squares  of  the  actual  velocities  of  the  molecules,  in  feet 
per  second  at  493.2°  F.  above  absolute  zero,  hydrogen  6282,  oxygen  1572,  carbonic  oxide 
1276,  carbonic  acid  1570.  Phil.  Mag.,  1873,  p.  68.  Our  equation  (4)  gives  for  air  1593. 

t  Phil.  Mag.,  1855  [4]  IX.,  p.  37. 


THE   LUMINIFEKOUS   ETHER.  375 

what  exceeds  the  tension  of  the  most  perfect  vacuum  yet  produced  by 
artificial  means,  so  far  as  we  are  informed.  Crookes  produced  a  vacuum 
of  .02  millionth  of  an  atmosphere  *  without  reaching  the  limit  of  the 
capacity  of  the  pumps  ;  and  Professor  Rood  produced  one  of  5-5^75  ffnnnr 
of  an  atmosphere  f  without  passing  the  limit  of  action  of  his  apparatus. 
The  latter  gives  a  pressure  per  square  foot  of 

14  7  X  144 

390000000"  =  Tr°1°1!TJ  of  a  P°und-     This>  in  round  numbers,  is  140  times 

the  value  given  in  equation  (11).  '  Even  at  this  great  rarity  of  the  atmos- 
phere, the  quantity  of  matter  in  a  cubic  foot  of  the  air  would  be  some 
200  million  million  times  the  quantity  in  a  cubic  foot  of  the  ether — such 
is  the  exceeding  levity  of  the  ether. 

Admitting  that  the  ether  is  subject  to  attraction  according  to  the 
Newtonian  law,  and  of  compression  according  to  the  law  of  Mariotte, 
we  propose  to  find  the  relation  between  tlie  density  of  the  ether  at  the  surface 
of  an  attracting  sphere  and  that  at  any  other  point  in  space,  providing 
that  the  sphere  be  cold  and  the  only  attracting  body,  and  the  gas  con- 
sidered the  only  one  involved. 

Let  oa>  <?o,  w0  be  respectively  the  density,  elasticity  and  weight  of  a 
unit  of  the  medium,  whether  ether,  air,  or  any  other  gas  at  the  surface 
of  the  sphere  ;  rf,  e,  w,  the  corresponding  quantities  at  a  distance  z  from 
the  surface  of  the  sphere  ;  r  the  radius  of  the  sphere,  ga  the  acceleration 
due  to  gravity  at  its  surface,  and  g  that  at  distance  r  -\- z  from  the  centre 
of  the  sphere.  Then 


and 


(12) 

But 

de=  —  wdz-  —  gSdz  (13) 

d  e._       g0  do          r2 

Integrating  between  e  and  e0,  z  and  o  we  have 

*  On  the  Viscosity  of  Gases  at  High  Exhaustion,  by  William  Crookes,  F.R.S., 
Phil.  Trans.  Roy.  Soc.,  Part  II.  (1881),  p.  400  :  "  Going  up  to  an  exhaustion  of  .02  mill- 
ionth of  an  atmosphere,  the  highest  point  to  which  I  have  carried  the  measurements, 
although  by  no  means  the  highest  exhaustion  of  which  the  pump  is  capable." 

t  Jovrn  of  Arts  and  Science,  1881,  Vol.  XXII.,  p.  90. 


376  APPENDIX   I. 

<7o  So        rz 


(14) 

(15) 

Neglecting  the  attraction  of  the  earth  for  the  ether,  and  considering 
the  sun  as  the  only  attracting  body,  we  have  g0  at  the  sun  28.6  X  32.2, 
and  at  the  earth,  z  =  210  r,  r  =  441,000  miles,  the  sun's  radius; 
6  =  •&  x  10-",  equation  (9),  and  e.=  -&  X  10- 6 ;  and  these,  in  (14) 
and  (15),  give 


-,..35*515  newly,  (16) 

and 


nearly,  (iff) 

for  the  tension  and  density  of  the  ether  at  the  surface  of  the  sun  under 
the  conditions  imposed.  But  the  millionth  root  of  c  is  practically  unity  ; 
hence  the  elasticity  and  density  at  the  sun  is  practically  the  same  as  at 
the  earth. 

Now,  starting  at  the  sun  with  this  result,  and  finding  the  density  at  a 
distance  z  from  it,  then  making  z  infinite,  we  shall  get  about  the  995,000 
root  of  c,  the  value  of  which  is  also  sensibly  equal  to  unity  ;  hence  the 
density  at  infinity  would  be  sensibly  the  same  as  at  the  surface  of  the 
sun,  the  difference  in  the  densities  at  the  sun  and  at  infinity  being  less 
than  TffBihnro  part  of  that  at  the  sun.  In  order  to  make  the  density  vary 
sensibly  with  the  distance,  the  attraction  of  the  central  body  must  be 
something  like  a  million  times  as  great  as  that  of  the  sun,  or  have  a 
diameter  a  million  times  as  large  ;  but  there  is  no  such  known  body, 
therefore  the  density  and  tension  of  the  ether  may  be  considered  uniform 
throughout  space.  Such  has  been  our  conception  of  it,  and  it  is  an  agree- 
able surprise  to  find  it  so  fully  confirmed  by  analysis. 

If  the  density  were  uniform,  the  weight  of  a  given  volume  of  it  would 
vary  as  the  force  of  gravity.  At  the  surface  of  the  sun  a  cubic  foot 
would  weigh  [equation  (10)  multiplied  by  28.  6,  or]  57  X  10  -";  hence, 
for  a  height  h  it  would  weigh 

57        r  h  n  ~ 


which  for  h  =  oo  becomes  ^  of  a  pound,  which  is  the  pressure  upon  a 
square  foot  of  the  sun  of  a  column  of  infinite  height  under  the  conditions 


THE  LUMINIFEROUS  ETHEE.  377 

imposed.  This  would  compress  the  first  foot  of  the  column  about 
iooiooo  °f  its  length,  and  would  cause  a  corresponding  increase  in  the 
density,  the  value  of  which,  after  this  compression,  will  be  found  by 
multiplying  the  value  given  in  equation  (9)  by  to^Wo^j.  which  will  leave 
the  result  sensibly  the  same  as  before.  Hence,  from  this  standpoint, 
we  again  conclude  that  the  density  of  the  ether  may  be  considered  as 
sensibly  uniform  throughout  space,  providing  its  temperature  be  essen- 
tially uniform. 

If  we  assume  that  the  law  of  the  resistance  by  which  the  ether  opposes 
the  motion  of  a  body  varies  as  the  square  of  the  velocity  of  the  body,  we 
are  still  unable  to  assign  the  coefficient  which  will  give  the  numerical 
value  ;  but  it  is  safe  to  assume  that  the  entire  mass  of  the  ether  occupy- 
ing the  path  of  a  body  moving  through  it,  will  not  have  a  velocity  im- 
parted to  it  exceeding  that  of  the  body  ;  but,  to  be  on  the  safe  side,  we 
will  assume  that  it  imparts  a  velocity  equal  to  itself.  The  energy  thus 
imparted  will  be  lost  to  the  body.  To  simplify  the  case,  consider  a 
planet  moving  in  a  circular  orbit :  r  the  radius  of  the  planet,  d  its  dis- 
tance from  the  sun,  D  its  specific  gravity  compared  with  water  as  unity, 
TI  the  velocity  in  its  orbit  ;  then  the  mass  of  ether  occupying  the  place 
of  the  planet  during  one  revolution  about  the  sun  will  be,  using 
equation  (9), 


35  X  1024 

which,  multiplied  by  £  «,2,  will  give  the  energy  imparted  to  it.     The 
kinetic  energy  of  a  planet,  neglecting  its  rotation,  will  be 


Dividing  the  former,  after  multiplying  it  by  \  v^,  by  the  latter,  gives 

1    ___  1  (18) 

7  X  10i4    r  D 

for  the  fraction  of  the  energy  lost  during  one  revolution  about  the  sun. 
Applying  this  to  the  earth,  we  have 

d-^-rD  =  93000000  -f  3912  X  5i  =  43000, 
and  (18)  becomes 

(19) 


for  the  fraction  of  the  energy  lost  in  one  year  ;  and  hence  at  this  rate 
would  require  more  than  1,666,000  trillion  (1,666,000,000,000,000,000,000) 
years  to  bring  it  to  rest. 

Equation  (18)  is  not  applicable  to  the  resistance  offered  to  a  comet,  on 
account  of  the  elongated  orbit  of  the  latter  ;  but  some  idea  of  the  effect 
of  the  resistance  of  the  ether  to  the  movement  of  a  comet  may  be  found 


378  'APPENDIX  i. 

by  considering  what  it  would  be  if  the  orbit  were  circular,  having  for  its 
radius  the  perihelion  distance.  According  to  Professor  Morrison,  the 
perihelion  distance  of  the  great  comet  (6),  1882,*  was  716200  miles,  its 
aphelion  distance  will  be  5,000,000,000  miles:  the  diameter  of  its  nucleus 
shortly  before  disappearing  on  the  solar  disk  was  7600  miles,  the  \eloeit\- 
at  perihelion  295  miles  per  second,  and  at  aphelion  75  feet  per  second. 
But  little  is  known  in  regard  to  the  density  of  comets  ;  but.  to  be  on  the 
safe  side,  we  will  assume  it  as  YOU?  that  of  water.  This  data  will  reduce 
(18)  to  13  X  10— 1S  for  the  fraction  of  energy  lost  during  one  of  its  revolu- 
tions about  the  sun  ;  and  as  it  would  make  a  revolution  in,  say,  20  hours. 
it  would  lose  in  one  of  our  years  about  57  X  10  ~ 16  of  its  energy,  ///  "•/  if// 
Tutt-  it  would  go  on  for  170  trillions  of  years.  Similarly,  at  its  aphelion 
its  rate  of  loss  would  be  less  than  \  X  10 ~ u  of  its  energy  in  more  than 
2000  years — the  time  of  one  revolution  in  its  orbit. 

The  most  careful  observations  and  calculations  have  failed  to  detect 
any  effect  due  to  the  resistance  of  matter  in  space  ;  and  the  alxjve  analysis 
shows  that,  within  historic  times,  it  has  in  any  case  scarcely  amounted  to 
an  infinitesimal,  certainly  not  sufficient  to  be  measured.  And  when  we 
consider  that  our  assumptions  have  been  very  largely  on  the  unfavorable 
side,  and,  further,  that  the  energy  imparted  to  the  ether  may  partly,  at 
lea-t.  he  restored  to  the  body,  we  assume  that  its  resistance  never  can  be 
measured.  Laplace,  when  he  found  that  the  force  of  gravitation,  if 
propagated  by  an  elastic  medium,  must  have  a  velocity  exceeding  100 
million  times  that  of  light,  concluded  that  astronomers  might  continue 
to  consider  its  action  as  instantaneous  (Mecanique  Celeste,  B.  X..  cli.  s, 
•p.  22.  1KI35)  ;  so  may  we,  with  nearly  as  much  confidence,  continue  to 
rounder  the  resistance  of  the  ether  as  nil. 

Equation  (6)  gives 

--        6.6  (186800 


-  2  X  32.2  X  1.4  X  772  ~ 

from  which  the  specific  heat  of  the  ether  may  be  found  if  its  temperature 
were  known.  M.  Fourier,  the  first  to  assign  a  value  to  tlie  term*  rut  un 
ofspnce,  assumed  it  to  be  somewhat  inferior  to  the  temperature  at  the 
poles  of  the  earth  or  about  50°  C.  to  60°  C.  below  zero.f  M.  Pouillet, 
considering  the  atmosphere  as  a  diathermanous  medium,  capable  of 
absorbing  in  different  degrees  the  radiant  heat  from  the  sun  and  the  dark 
heat  from  the  earth,  deduced  for  the  heat  of  space — or,  as  he  and  Fourier 
called  it,  the  stellar  heat— approximately,— 142"  C.\  (-  287°  F.),  which 

*  Monthly  Notices  of  tfie  Royal  Astronomical  Society,  Vol.  XL1V.,  2,  p.  54. 

t  Ann.  der  Ckemte,  Tome  XVII ,  p.  155. 

$  Complex  Rendus,  1838,  Vol.  VII.,  p.  61.    Ponillet's  formula  is 


THE   LUMINIFEROUS   ETHER.  .     379 

is  about  174°  F.  above  absolute  zero.  It  is  well  known  that  Pouillet's 
data  were  imperfect,  several  important  elements  being  neglected,  notably 
that  of  the  humidity  of  the  air  ;  still,  it  is  not  only  the  first,  but,  so  far 
as  we  know,  the  only  attempt  to  formulate  this  relation.  It  served  to 
show  what  has  since  been  indicated  by  more  direct  experiments,  that  the 
temperature  of  space  is  very  low.  The  delicate  experiments  of  Professor 
Langley,  before  referred  to,  show  a  great  difference  in  the  degree  of 
absorption  by  our  atmosphere  of  different  wave-lengths.  The  mean  of 
the  values  for  nine  different  wave-lengths,  treated  by  M.  Pouillet's 
formula,  gives  139°  F.  above  absolute  zero,  and  the  smallest  value  of 
absorption,  which  was  for  the  infra-red,  gives  only  71°  F.  above  absolute 
zero  for  the  heat  of  space. 

*  The  heat  of  space  may  be  considered  as  composed  of  three  parts  : 
(T)  stellar  heat,  (2)  the  heat  contained  in  the  dark  matter  of  space,  (3)  the 
essential  heat  of  the  ether. 

1.  By  the  stellar  heat  we  mean  the  heat  received  directly  from  the 
stars.     It  is  a  matter  of  easy  calculation  that,  if  the  50,000,000  of  stars 
supposed  to  be  visible  with  the  most  powerful  telescopes  were  all  at  the 
distance  of  the  nearest  fixed  star  (a,  Centauri).  or  221,000  astronomical 
units  from  the  earth,  and  if  each  radiated  the  same  amount  of  heat  as 
our  sun,  the  intensity  varying  as  the  inverse  squares  of  the  distances,  the 
earth  would  receive  from  them  all  less  than  -^^  as  much  heat  as  it  now 
receives  from  the  sun.     And  when  we  consider  that  only  a  very  few  stars 
are  within  measurable  distances,  and  that  the  remote  ones  may  be,  when 
compared  with  these,  well-nigh  infinitely  distant,  it  is  evident  that  the 
amount  of  heat  received  from  the  stars  is  insignificant,   and  may  be 
discarded  at  the  earth. 

2.  It  is  certain  that  there  is  a  large  amount  of  dark  matter  in  space, 
since  the  meteoric  dust  and  meteorites  must  come  from  beyond  our 
atmosphere.     The  zodiacal  light  is  supposed  to  be  an  evidence  of  meteoric 
matter  between  the  earth  and  sun.     The  tails  of  comets  are  visible  by 
some  action  of  light  upon  some  kind  of  matter.     Matter  in  space  not 
exposed  to  the  rays  of  the  sun  will  be  at  about  the  same  temperature  a:j 
the  ether  ;  but  if  in  the  rays  of  the  sun  and  destitute  of  an  atmosphere 
at  the  distance  of  the  earth  from  the  sun,  its  temperature  would  be  very 
low.     If  present  laws  can  be  extended  so  far,  and  the  earth  were  without 
an  atmosphere,  and  the  heat  received  were  not  conducted  away,  it  has 
been  computed  that  the  mean  temperature  at  the  equator  would  be  about 

in  which  V  =  the  absorptive  power  by  the  atmosphere  of  the  sun's  heat, 

b  =  the  absorptive  power  of  terrestrial  heat, 

t'  —  the  temperature  of  the  stellar  heat, 

a  =  1.0077. 

If  b  =  1,  its  maximum,  V  =  0.2,  we  find  t'  =  —  235°  C.  (—  391°  F.),  or  71°  F.  above 
absolute  zero. 


380  APPENDIX   I. 

—  70°  C.  (-  94°  F.) ;  and  at  the  poles  —  221°  C.,*  or  114°  F.  above  abso- 
lute zero.  The  last  result  is  obtained  on  the  supposition  that  the  poles 
receive  heat  directly  from  the  sun  a  part  of  the  year  ;  it  is  further  shown 
that  if  the  poles  were  never  exposed  to  the  rays  of  the  sun,  the  tempera- 
ture would  fall  to  that  of  the  ether  of  space.  But  the  data  are  not  uni- 
form, and  there  is  too  large  an  extension  of  empirical  formula  to  satisfy 
one  that  the  above  numerical  results  are  reliable  :  still  they  point  more 
and  more  strongly  to  a  temperature  not  many  degrees  above  absolute 
zero.  » 

3.  By  the  essential  heat  of  the  ether  we  mean  the  temperature  which 
would  be  indicated  by  a  thermometer  graduated  from  absolute  zero  in  a 
room  located  in  space  beyond  our  atmosphere,  whose  walls  were  imper- 
vious to  the  passage  of  external  heat.  It  is  the  heat  due  to  the  self- 
agitated  ether,  just  as  air  has  a  temperature  when  not  exposed  to  the 
rays  of  the  sun.  If  the  ether  be  perfectly  diathermauous  to  the  sun's 
rays,  it  will  receive  no  heat,  on  account  of  the  heat  of  the  sun  flowing 
through  it,  though  it  may  be  heated  from  other  sources.  As  direct 
evidence  of  an  extremely  low  temperature  of  space,  we  cite  the  facts  in 
regard  to  the  meteorite  which  fell  at  Dharmsalla,  India,  July  14th,  1860.  f 
"  The  most  remarkable  thing  about  it  was,  while  the  mass  had  been  in- 
flamed and  melted  at  the  surface,  the  fragments  gathered  immediately 
after  the  fall  and  held  for  an  instant  were  so  cold  that  the  fingers  irere 
chilled.  This  extraordinary  assertion,  which  is  contained  in  the  report 
with  no  expression  of  doubt,  indicates  that  the  mass  of  the  meteorite  re- 
tained in  its  interior  the  intense  cold  of  the  interplanetary  space,  while 
the  surface  was  ignited  in  passing  through  the  terrestrial  atmosphere. " 
Since  this  body  had  been  exposed  to  the  rays  of  the  sun,  its  temperature 
must  have  exceeded  that  of  the  space  through  which  it  passed,  as  well 
as  been  warmed  by  the  heat  developed  at  its  surface,  from  which  it  may 
be  inferred  that  it  had  been  intensely  cold.  Direct  investigations,  given 
above,  indicate  that  this  temperature  is  less  than  200°  F.  above  absolute 
zero  ;  and  we  cannot  assert  that  it  is  not  less  than  100°  F.  above,  or  even 
much  less. 

But,  however  low  be  the  temperature  of  the  ether,  it  cannot  be  abso- 
lutely cold,  or,  in  other  words,  it  must  have  a  temperature  above  abso- 
lute zero,  for  otherwise  it  would  be  destitute  tof  elasticity,  and  hence 
incapable  of  transmitting  a  wave.  This  is  shown  by  eliminating  F  be- 
tween equations  (2)  and  (6),  giving 

(21) 


«  Professional  Papers  of  the  Signal  Service  17.  S.  A.,  Washington,  D.  C.,  1884,  No. 
XII.,  p.  54. 
t  Comptes  Rendus,  1861,  Tome  LIII.,  p.  1018. 


THE   LUMINIFEROUS   ETHER.  381 

In  which  if  r  =  0,  e  will  be  zero,  all  the  other  factors  being  nnite,  and  if 
e  =  0,  then  V  =  0  in  (2).  Indeed,  this  principle  is  so  well  recognized  in 
physics,  that  a  proof  in  this  place  seems  superfluous.  Being  unable,  in 
the  present  state  of  our  knowledge,  to  do  more  than  assign  the  probable 
superior  limit  of  the  temperature,  we  will,  for  the  purposes  of  this 
analysis,  assume  r  =  20°  F.,  absolute,  being  confident  that  the  actual 
value  is  between  -^  of  and  10  times  this  value.  This  value  in  equation 
(20)  gives 

c  =  46  X  10U  =  4,600,000,000,000  (22) 

for  the  specific  heat  of  the  ether,  that  of  water  being  unity.  This  num- 
ber so  vastly — we  might  say  infinitely — exceeds  that  for  any  known  gas, 
as  to  justify  one,  at  first  thought,  in  looking  with  suspicion  upon  the 
applicability  of  the  above  analysis  to  this  medium.  Assumptions  in  re- 
gard to  the  absolute  temperature  will  scarcely  improve  the  appearance  of 
this  number.  If  it  be  assumed  that  the  absolute  temperature  be  only 
one  degree,  the  number  in  equation  (22)  would*  be  only  twenty  times  as 
large;  and  if  the  absolute  temperature  be  assumed  at  1000000°  F. ,  the  result- 
ing specific  heat  would  still  be  more  than  a  million  times  as  large  as  for 
hydrogen.  A  few  considerations  of  other  properties  of  the  ether  may  aid 
one  in  being  reconciled  to  this  seemingly  paradoxical  result.  Is  the  result 
any  more  incredible  than  the  fact,  generally  admitted,  that  every  particle 
of  the  ether,  in  transmitting  a  wave  of  light,  continually  makes  590000- 
000000000  (6  X  1014  nearly)  complete  cycles  of  movements  every  second, 
for  a  wave-length  of  ^^inr  of  an  inch  ?  The  number  of  such  complete 
movements  in  air  for  the  fundamental  c  is  only  264  ;  and  hence  the  ratio 
of  the  former  to  the  latter  of  these  numbers  is  nearly  2  X  1012.  The  ratio 
of  the  specific  heat  given  in  (22)  to  that  of  hydrogen  is  nearly  1J  X  10", 
which  is  not  so  different  from  that  just  given  for  the  ratio  of  cyclical 
movements  in  a  second  of  the  ether  and  air.  The  velocity  of  sound  in 
air  at  493°  F.  above  absolute  zero  is  about  1090  feet  per  second  ;  but  if 
the  temperature  could  be  reduced  to  20°  F.,  absolute,  the  law  being  ex- 
tended so  far,  the  velocity  would  be  only 

V=mOi/^  =  217  feet; 

r      4Uo 

but  the  velocity  of  light  is  982,000,000  feet  per  second,  a  number  about  4£ 
million  times  the  former,  and  near  a  million  times  that  of  the  velocity 
in  air  under  ordinary  conditions.  The  ratio  of  the  mass  of  air  in  a  cubic 
foot  at  sea-level  to  that  of  a  cubic  foot  of  the  ether  as  computed,  far 
exceeds  any  of  these  ratios.  The  fact  is,  the  known  qualities  of  the 
ether  in  transmitting  light  and  heat  so  far  transcend  those  of  any  known 
terrestrial  substance,  that  we  might  anticipate  the  fact  that,  in  regard  to 
magnitude,  all  its  properties  will  be  extremely  exceptional  when  com- 
pared with  such  substances.  We  must  accept  substantially  the  number 


382  APPENDIX   I. 

in  equation  (22),  or  subject  this  medium  to  different  laws  than  those  of 
gases. 

We  may  deduce  this  result  by  another  process  ;  thus,  since  the  *}><•(  -itic 
heats  of  different  gases  arc  as  the  squares  of  the  wave-  velocities  in  the 
respective  substances,  the  other  elements  being  the  same,  if  the  specific 
heat  of  air  be  0.23,  we  should  have  for  the  specific  heat  of  the  ether 


as  before.  The  correct  value  of  the  specific  heat  of  air,  0.2375,  would 
give  over  47  X  10",  and  nearly  48  X  1011  ;  but  these  differences  are  quite 
immaterial  in  this  connection,  the  object  being  to  check  the  former  result. 
On  the  other  hand,  in  order  that  common  air  might  be  able  to  transmit 
a  wave  with  the  known  velocity  of  light,  its  sped  lie  heal  being  taken  con- 
stantly at  0.23,  its  temperature  would  be,  according  to  equation  (20), 


=  4  X  10U  degrees  F.  (=  400,000,000,000,000°  F.). 


If  the  sun  were  composed  of  a  substance  having  such  specific  heat,  it 
could  radiate  heat  at  its  present  rate  for  more  than,  a  hundred  millions  of 
centuries  without  its  temperature  being  reduced  1°  F.,  exclusive  of  any 
supply  from  external  sources,  or  from  a  contraction  of  its  volume.  We 
know  only  such  substances  in  the  sun  as  we  are  able  to  experiment  with 
in  the  laboratory;  and  if  there  be  an  exceptional  substance  in  it,  we  have 
no  means  at  present  of  determining  its  physical  properties.  It  is,  more- 
over, a  question  whether  the  ether  constitutes  an  essential  part  of  bodies. 
We  conceive  of  it  only  as  the  great  agent  for  transmitting  light  and  heat 
throughout  the  universe. 

On  account  of  the  enormous  value  of  the  specific  heat,  it  will  require 
an  inconceivably  large  amount  of  heat  (mechanically  measured)  to  in- 
crease the  temperature  of  one  pound  of  it  perceptibly.  Thus,  it  heat 
from  the  sun,  by  passing  through  a  pound  of  water  at  the  earth,  would 
raise  the  temperature  100°  F.  and  maintain  it  at,  say,  600°  F.,  absolute, 
it  would,  under  similar  conditions,  raise  the  temperature  of  one  pound  of 
the  ether,  if  its  power  of  absorption  be  the  same  as  that  of  water, 
iauouooooao  of  a  degree. 

The  distance  of  the  earth  from  the  sun  being  210  times  the  radius  of 
the  latter,  the  amount  of  heat  passing  a  square  foot  of  spherical  surface 
at  the  sun  will  be  about  45000  times  the  heat  received  on  a  square  foot  at 
the  earth  normally  exposed  to  its  rays,  so  that,  under  the  conditions 
imposed,  the  temperature  would  not  be  a  billionth  of  a  degree  F.  higher 
at  the  sun  than  at  the  earth.  This,  then,  is  a  condition  favorable  to  a 
sensibly  uniform  temperature,  even  if  heated  by  the  sun's  rays.  We  are 
now  inclined  to  admit  that  the  ether  is  not  perfectly  diathermanous  to 


THE   LUMINiFEROUS   ETHEK.  383 

the  sun's  rays,  but  that  its  temperature,  however  small,  may  be  due 
directly  to  the  absorption  of  the  heat  of  central  suns ;  for  we  begin  to 
•  realize  the  fact  that  the  ether  may  possess  many  of  the  qualities  of  gases, 
such  as  a  molecular  constitution,  and  hence  also  mass,  elasticity,  specific 
heat,  compressibility,  and  expansibility,  although  the  magnitude  of  these 
properties  is  anomalous.  We  have  already  considered  its  compressibility 
at  the  surface  of  the  sun,  due  to  the  weight  of  an  infinite  column,  and 
found  it  to  be  exceedingly  small  ;  now,  it  may  be  possible  that  the  expan- 
sion due  to  the  excess  of  temperature  of  a  small  fraction  of  one  degree  at 
the  surface  of  the  sun  over  that  at  remote  distances  will  diminish  the 
density  as  much,  or  about  as  much,  as  pressure  increased  it,  thereby 
making  the  density  even  more  exactly  uniform  than  it  otherwise  would 
be.  According  to  what  we  know  of  refraction,  it  is  impossible  for  a  ray 
of  light  to  be  refracted  in  passing  through  the  ether  only — at  least,  not 
by  a  measurable  amount ;  for  not  only  are  the  density  and  elasticity 
practically  uniform,  but  their  ratio  is,  if  possible,  even  more  constant  as 
shown  by  equations  (16)  and  (16').  But  the  freedom  of  the  ether  mole- 
cules may  be  constrained,  or  their  velocity  impeded,  by  their  entangle- 
ment with  gross  matter,  such  as  the  gases  and  transparent  solids  ;  in 
which  case  refraction  may  be  produced  in  a  ray  passing  obliquely  through 
strata  of  varying  densities.*  Neither  is  it  believed  that  the  ether  does, 
or  can,  reflect  light  ;  for  if  it  did,  the  entire  sky  would  be  more  nearly 
luminous.  The  rays  in  free  space  move  in  right  lines. 

The  masses  of  the  molecules  in  different  gases  being  inversely  as  their 
specific  heats,  and  as  the  specific  heat  of  hydrogen  is  3.4,  and  the  com- 
puted mass  of  one  of  its  molecules  { J  X  10  ~ 29  f  of  a  pound,  we  have  for 

*  Professor  Michaelson  concludes  from  his  experiments  that  the  luminiferous  ether 
has  no  perceptible  motion  in  reference  to  the  earth,  in  other  words,  it  is  at  the  surface 
of  the  earth  carried  along  with  the  earth  the  same  as  the  atmosphere.  (Paper  read  at 
the  meeting  of  the  American  Association  for  the  Advancement  of  Science,  1887.) 

t  Stoney  concludes  that  "  it  is  therefore  probable  that  there  are  not  fewer  than  some- 
thing like  a  unit  eighteen  (1018)  of  molecules  in  a  cubic  millimeter  of  a  gas  at  ordinary 
temperature  and  pressure"  (PhU.  Mag.,  1868  [4]  XXXVI.,  p.  141).  According  to  the 
Kinetic  theory,  the  number  of  molecules  in  a  given  volume  under  the  same  pressure  and 
temperature  is  the  same  for  all  gases.  The  weight  of  a  cubic  foot  of  hydrogen  at  the 
temperature  of  melting  ice  and  under  constant  pressure  being  0.005592  of  a  pound,  and 
as  a  cubic  foot  equals  28,315,000  cubic  millimeters,  the  probable  mass  of  a  molecule  of 
hydrogen  will  be 

0.005592 11 

32.2  X  28315000  X  1035  ~~  18  X  10" 

Maxwell  gives  -^5  of  a  gramme  =  - — £— •  lb.,  which  is  about  3/5 the  value  given  above 

(PhU.  Mag.,  1873  [4],  XLVL,  p.  468). 

The  difference  in  these  results  arises  chiefly  from  the  calculated  number  of  molecules 
in  a  cubic  foot  of  gas  under  ordinary  conditions.  Thomson  gives  as  the  approximate 


384  APPENDIX   I. 

the  computed  mass  of  a  molecule  of  the  luminiferous  ether, 

3.4  _1  __  {23) 

22  X  1040 

The  mass  of  a  cubic  foot  of  the  ether,  equation  (to),  divided  by  the  mass 
of  a  molecule,  gives  the  number  of  molecules  in  a  cubic  foot,  which 
will  be 


which  call  101*.  This  number,  though  large,  is  greatly  exceeded  by  the 
estimated  number  of  molecules  in  a  cubic  foot  of  air  under  standard  con- 
ditions, which,  according  to  Thomson,  does  not  exceed  17  X  10",  a 
number  nearly  17,000,000,000  times  as  large  as  that  in  equation  (24)  ;  and 
yet,  at  moderate  heights,  the  number  of  molecules  in  a  given  volume  of 
air  will  be  less  than  that  of  the  ether. 

Assuming  that  air  is  compressed  according  to  Boyle's  law,  and  is  sub- 
jected to  the  attraction  of  the  earth,  equation  (15)  will  give  the  law  of 
the  decrease  of  the  density.  Taking  the  density  of  air  at  sea-level  at  ^ 
of  a  pound  per  cubic  foot,  <?„  =  14.7  Ibs.  per  square  inch,  r  =  20687000 
feet,  equation  (15)  becomes 

«J  =  TATX10-M5FT^.  (25) 

Itz  —  oo,<J  =  T^XlO-3*8,  which  would  be  the  limit  of  the  density, 
and  it  is  a  novel  coincidence  that  this  limit  is  nearly  identical  with  the 
value  found  for  the  density  at  the  height  of  one  radius  of  the  earth  ac- 
cording to  the  ordinary  exponential  law,  wherein  gravity  is  considered 
uniform.* 

If  the  number  of  the  molecules  in  a  cubic  foot  follows  the  same  law, 
then  at  the  height  z  there  will  be 

17  X  10-mrT^  +  25  (26) 

probable  number  17  X  10**,  which  is  about  3/5  the  value  given  by  Stoney.    Thomson's 
value  would  make  the  mass  of  a  molecule  of  ether  about  —  X  10~«°  of  a  pound,  which 
is  not  much  different  from  that  found  above. 
*  The  ordinary  exponential  law  results  from  dropping  -  compared  with  unity  in  equa- 

tion (15),  giving 

z  zft.  g  miles 

6  =  d0  e-2^1  =  (Jo  10  ~««87  =  -fa  X  10  ~   H.44  . 
f  n  the  last  of  which,  if  z  =  3956,  the  exponent  becomes  345. 


THE   LUMINIFEROUS    ETHER. 


385 


molecules  per  cubic  foot.     Similarly,  the  value  of  the  length  of  the  mean 
free  path  would  be  * 

2  X  10S45r+  2~£  (27) 

By  means  of  these  values,  the  following  table  may  be  formed. 


Height. 

Density  or  ten- 
sion, that  at 
the  earth 
being  unity. 

Number  of 
molecules  in  a 
cubic  foot. 

Length  of  the 
mean 
free  path. 

Fractional 
parts  of 
earth's  radius. 

i 
Approximate 
in  miles. 

0 

0 

1 

17  X  1025 

2  X  10  -  6    inch. 

W 

50 

10  -  4.3 

17  X  1020'7 

2  X  10  -1'7    " 

16  6 

O  A 

aV 

100 

10-8.4 

17  X  10 

2  X  10 

A 

200 

10-16.4 

17  X  108'6 

792000  miles. 

A 

282 

10-23 

17  X  10* 

31  X  1011 

A 

395 

10-31 

17  XlO~6 

31  X  1019 

i 

800 

10  -57 

17X10-32 

31  X  1045 

i 

3956 

10  -  17* 

17  X  10  -  147 

31  X  10160       " 

2 

7912 

10-230 

17  XlO"205 

31  X    I218       " 

oo 

00 

10  —  345 

17  XlO-32031  X  10333       " 

The  numbers  in  the  third  column  multiplied  by  ffa  will  give  the 
density  (or  mass  per  cubic  foot)  at  the  respective  altitudes  ;  and  the  same 
numbers  multiplied  by  15  (or,  more  accurately,  14. 7)  will  give  the  tension 
per  square  inch.  According  to  this  law,  at  an  elevation  of  300  miles  the 
density  of  the  atmosphere  will  be  somewhat  less  than  the  density  of  the 
ether  as  given  by  equation  (9). 

To  find  the  height  at  which  the  tension  of  the  atmosphere,  according 
to  the  above  law,  will  be  the  same  as  that  of  the  ether,  we  have,  by  means 
of  equations  (11)  and  (25),  substituting  in  the  latter  2116  for  ifa, 


which  solved  gives 


2116  X  lO"345^  =  —. 


*  PhU.  Mag.,  1873  [4],  XLV1.,  p.  468. 


886  APPENDIX   I. 

BO  that  at  the  height  of  127  miles  the  tension  would  be  less  than  that  of 
the  ether,  the  temperature  being  uniform. 

The  mean  free  path,  according  to  the  above  law,  in  which  gravity 
varies  as  the  inverse  squares  is  less,  and  for  great  heights  much  less,  ilian 
would  be;  found  according  to  the  ordinary  exponential  law.  Thus 
Crookes  states  that  the  mean  free  path  of  a  molecule  at  the  height  of  200 
miles  is  about  10000000  miles  ;  *  but  according  to  the  above  law  it 
Ix-eomes  about  792000  miles. 

If  a  cubic  inch  of  air  at  sea-level  were  carried  to  the  height  of  I  the 
radius  of  the  earth,  and  then  allowed  to  expand  freely,  so  as  to  become 
of  the  computed  density  of  the  atmosphere  at  that  point,  it  would  fill  a 
space  of  4  X  1058-1-  cubic  miles,  or  a  sphere  whose  radius  is  2,898,000,000 
miles,  which  is  nearly  equal  to  the  distance  of  the  planet  Neptune  from 
the  sun  ;  and  there  would  be  less  than  one  molecule  to  the  mile.  Such 
are  some  of  the  results  of  extending  a  law  to  extreme  cases  regard h •>--  «l' 
physical  limitations  or  of  the  imperfection  of  the  data  on  which  it  is 
founded.  For  instance,  a  uniform  temperature  is  assumed,  and,  im- 
pliedly,  an  unlimited  divisibility  of  the  molecules.  The  latter  is  neces- 
sary in  order  to  maintain  a  law  of  continuity.  But  modern  investiga- 
tions show  that  not  only  air,  but  all  the  gases,  are  composed  of  molecules 
of  definite  magnitudes  whose  dimensions  can  be  approximately  deter- 
mined ;  and  hence  if  there  IK?  only  a  few  molecules  in  a  cubic  foot,  and 
much  less  if  there  be  but  one  molecule  in  a  cubic  mile,  it  cannot  be 
claimed  that  the  gas  will  be  governed  by  the  same  laws  as  at  the  surface 
of  the  earth. 

We  conclude,  then,  that  a  medium  whose  density  is  such  that  a  volume 
of  it  equal  to  about  twenty  volumes  of  the  earth  would  weigh  one  pound, 
and  whose  tension  is  such  that  the  pressure  on  a  square  mile  would  be 
about  one  pound,  and  whose  specific  heat  is  such  that  it  would  require 
as  much  heat  to  raise  the  temperature  of  one  pound  of  it  1°  F.  as  it 
would  to  raise  about  2,300,000,000  tons  of  water  the  same  amount,  will 
satisfy  the  requirements  of  nature  in  being  able  to  transmit  a  wave  of 
light  or  heat  186300  miles  per  second,  and  transmit  133  foot-pounds  of 
heat-energy  from  the  sun  to  the  earth,  each  second  per  square  foot  of 
surface  normally  exposed,  and  also  be  everywhere  practically  non-resist- 
ing and  sensibly  uniform  in  temperature,  density  and  elasticity.  This 
medium  we  call  the  Luminiferous  Ether. 

ADDENDA. 

Granting  that  the  temperature  of  the  ether,  however  low,  is  produced 
by  the  heat  from  central  suns  passing  through  it,  we  may  determine  the 
effect  upon  it  of  a  change  of  temperature  of  the  source  of  heat. 

*  Phil.  Trans.  Roy.  Soc.,  London,  1881,  Part  II.,  p.  389. 


THE   LUMKMl-KKOUS    ETHER.  c'.87 

The  law  for  perfect  gases  is  —  continuing  our  notation  — 

e  v  =  R  T  (36) 

where  R  is  e0  v0  -f  r0)  these  values  being  initial.  Since  ®  will  necessarily 
be  constant  we  see  that  e  will  vary  as  T,  where  r  is  the  temperature  of 
the  ether,  and  equation  (21)  becomes 

e 
—  •  -  =  constant, 


as  it  should,  since  the  mean  density  cannot  change,  the  volume  being 
constant.  This  equation  reveals  no  new  truth,  but  is  consistent  with  the 
conditions  which  we  anticipate  in  nature.  The  only  way  in  which  the 
density  can  change  by  a  diminution  of  elasticity  of  the  ether,  is  to  cause 
it  to  be  more  dense  near  the  attractive  bodies,  and  more  rare  in  space 
more  remote  from  them  ;  or,  in  other  words,  the  ether  would  not  be  so 
nearly  uniform  as  at  present. 

Assuming  the  density  as  uniform  while  the  elasticity  changes,  it 
appears  from  equation  (2)  that  the  velocity  of  light  through  it  will  vary 
as  the  square  root  of  the  elasticity.  Thus,  if  the  heat  of  our  sun  dimin- 
ishes so  as  to  become  one  fourth  as  intense  as  at  present,  and  if  the 
elasticity  of  the  ether  also  becomes  one  fourth  as  much  as  at  present, 
then  will  the  velocity  of  light  be  one  half  as  great  as  at  present. 


We  may  find  the  conditions  which  would  cause  a  gas  of  the  pressure 
of  our  atmosphere  at  sea-level  and  of  the  same  specific  heat,  to  be  as 
nearly  uniform  throughout  space  as  is  the  ether.  This  will  be  found 
with  sufficient  accuracy  for  our  purpose  by  finding  such  a  value  for  <5  as 

will  make  the  numerator  in  equation  (15),  -       ^^  ,  the  same  as  given  in 

(16),  where  ea  =  2116  the  tension  of  the  air  per  square  foot.     We  will 
find 


The  volumes  being  inversely  as  the  densities,  the  last  result  combined 
with  equation  (36)  shows  that  the  required  rarity  (or  density)  may  be 
secured  by  a  temperature  1015  times  that  of  the  present  temperature.  If 
the  absolute  temperature  be  500°  when  the  pressure  of  the  air  per  square 
foot  is  2000  pounds,  then  if  it  be  heated  to  something  like 

500,000,000,000,000°  F.,    . 

the  tension  would  be  nearly  uniform  throughout  space.  A  volume  of 
such  air  of  the  size  of  the  earth  would  weigh  less  than  jfo  of  a  pound  at 
a  place  where  g  =  32. 2. 


APPENDIX  II. 


SECOND   LAW   OF   THERMODYNAMICS. 

THE  second  law  of  thermodynamics  has,  by  different  writers,  been 
stated  in  a  variety  of  ways,  and,  apparently,  with  ideas  so  diverse  as  not 
to  cover  a  common  principle.  For  the  convenience  of  the  student  in 
considering  this  subject,  we  here  quote  some  of  the  expressions  which 
have  been  given  by  certain  authors. 

Maxwell,  in  his  Theory  of  Heat,  p.  153,  says,  "  Admitting  heat  to 
be  a  form  of  energy,  the  second  law  asserts  that  it  is  impossible,  by  the 
unaided  action  of  natural  processes,  to  transform  any  part  of  the  heat  of 
a  body  into  mechanical  work,  except  by  allowing  heat  to  pass  from  that 
body  into  another  at  a  lower  temperature.  Clausius,  who  first  stated  the 
principle  of  Carnot  in  a  manner  consistent  with  the  true  theory  of  heat, 
expresses  this  law  as  follows  : 

"  '  It  is  impossible  for  a  self -acting  machine,  unaided  by  any  external 
agency,  to  convert  heat  from  one  body  to  another  at  a  higher  temperature.' 

"  Thomson  gives  it  a  slightly  different  form  : 

"  '  It  is  impossible,  by  means  of  inanimate  material  agency ,  to  derive  me- 
chanical effect  from  any  portion  of  matter  by  cooling  it  below  the  tempera- 
ture of  the  coldest  of  surrounding  objects.'  "  The  last  quotation  may  be 
found  in  Phil.  Mag.,  1852,  IV.  ;  Thomson's  Mathematical  and  Physical 
Papers,  p.  179  ;  and  Clausius's  statement  on  p.  181. 

Clausius  considers  this  principle  as  "  a  new  fundamental  principle," 
and  states  it  thus  :  "  Heat  cannot  pass  from  a  colder  to  a  hotter  body 
without  compensation."  (Mechanical  Theory  of  Heat,  Browne's  transla- 
tion, p.  78.) 

It  appears,  so  far  as  we  can  judge,  that  Maxwell  has,  gratuitously, 
claimed  for  these  writers  the  above  statement  for  the  second  law  ;  for 
not  only  they,  but  Rankine  included,  consider  those  statements  as  ax- 
ioms. In  regard  to  Rankine's  views,  see  Miscellaneous  Scientific  Papers, 
p.  449  ;  Steam-Engine,  p.  224. 

There  would  be  a  certain  propriety  in  calling  this  the  second  law.  and  if 
necessary  establish  a  third,  for  it  is  the  first  principle  in  the  order  of  de- 
velopment involved  in  the  physical  operation  of  realizing  Carnot's  cycle, 
in  which  the  expansion  being  isothermal  requires  a  supply  of  heat  from 


390  APPENDIX   II. 

a  source,  and  experience  shows  that  the  temperature  of  the  source  must 
at  least  equal  that  of  the  working  substance,  and  in  reality  be  iufinitesi- 
mally  higher,  since  heat  from  a  colder  body  will  not  make  a  hot  body 
hotter.  But  the  question  is  not  what  might  have  been  the  second  law, 
but— what  is  it  ?  We  quote  from  Rankine  : 

"  The  internal  work  is  incapable  of  direct  measurement.  Here  it  is 
that  the  second  law  becomes  useful ;  for  it  informs  us  how  to  deduce  the 
whole  amount  of  work  done —internal  and  external— from  the  knowl- 
edge which  we  have  of  the  external  work.  That  law  is  capable  of  being 
stated  in  a  variety  of  forms,  expressed  in  different  ways,  although  virtu- 
ally equivalent  to  each  other.  The  most  convenient  form  for  the  present 
purpose  appears  to  be  the  following  : 

To  find  the  whole  work,  internal  find  external,  multiply  the  ab&olnt  /•  //-- 
perature  at  which  the  cJuinge  of  dimensions  takes  place  by  tJie  rait  /<•  /  <l<- 
gree  at  which  tlie  external  work  is  varied  by  a  small  variation  of  h  /«/« /v- 
ture."  (Rankine'3  Miscellaneous  Scientific  Papers,  p.  434  ;  The  Ei«jii<«  / , 
June  28,  1867.) 

This  is  substantially  the  statement  of  the  second  law  in  first  ed.,  p. 
33,  since  the  italicized  extract  just  given  is  an  expression  for  the  heat 
absorbed  during  an  isothermal  expansion.  The  form  in  the  text  was  not 
given  because  it  was  considered  the  ideally  best  statement  of  this  Ia\\ , 
but  because  it  had  proved  to  be  the  most  useful  form  for  class-room  in- 
struction which  the  author  had  tried,  and  had  the  above  sanction  of  Ran- 
kine. 

Rankine  gives  substantially  the  same  statement  in  different  places. 
(Papers,  pp.  309,  418,  427  ;  Steam-Engine,  p.  308,  Art.  244  ;  p.  309,  Art. 
245.) 

That  Rankine  recognized  Carnofs  principle  of  the  elementary  reversi- 
ble engine  as  the  second  law  is  shown  from  the  following  extract : 

"  The  law  of  efficiency  of  a  perfect  lieat  engine  nmy  be  stated  thus  :  If  the 
substance  (for  example,  air  or  water)  which  does  the  work  -in  a  perfect  limt 
engine  receives  all  the  heat  expended  at  one  fixed  temperature,  and  gir,x  <>nt 
aU  tlie  heat  which  remains  unconverted  into  icork  at  a  lower  fixed  t<  /////<  ra- 
ture,  tlie  fraction  of  the  whole  heat  expended  which  is  converted  into  exter- 
nal work  is  expressed  by  dividing  the  difference  between  tliose  tempt- r<it>ir<x 
by  the  higJier  of  them,  reckoned  from  the  absolute  zero.  3~ow,  this  is,  in 
fact,  tJie  second  law  of  thermodynamics  expressed  in  other  words."  (Wtu'tl'n- 
neom  Sc.  Papers,  p.  436  ;  The  Engineer,  June,  1867.)  Such  being  Ran- 
kine's  explicit  statement,  we  may  expect  to  find  this  principle  implied, 
if  not  expressed,  in  all  his  other  statements. 

One  of  the  most  condensed  and  obscure  statements  of  this  law  by  this 
author  is  in  his  work  on  the  Steam-Engine,  p.  306,  which  is, 


SECOND   LAW   OF   THERMODYNAMICS.  391 

"  If  the  total  actual  heat  of  a  homogeneous  and  uniformly  hot  substance 
be  conceived  to  be  divided  into  any  number  of  equal  parts,  the  effects  of  those 


The  obscurity  exists  chiefly  in  the  fact  that  several  principles  involved 
in  the  practical  application  of  the  law  are  not  stated  in  the  immediate 
context,  but  are  left  for  the  reader  to  infer.  It  would  be  difficult,  if  not 
impossible,  to  apply  it,  had  not  the  author  given  a  symbolic  representa- 
tion of  it.  He  says  :  "  Let  unity  of  weight  of  a  homogeneous  substance, 
possessing  the  actual  heat  Q,  undergo  any  indefinitely  small  change,  so 
as  to  perform  the  indefinitely  small  work  d  IT.  It  is  required  to  find  how 
much  of  this  work  is  performed  by  the  disappearance  of  heat.  Conceive 
Q  to  be  divided  into  an  indefinite  number  of  indefinitely  small  parts, 
each  of  which  is  6  Q.  (In  his  original  paper  he  used  d  Q.)  Each  of 
these  parts  will  cause  to  be  performed  the  quantity  of  work  represented 

b 


consequently,  the  work  performed  by  the  disappearance  of  heat  will  be 

d(d  U)." 
~ 


d  Q 

The  reduction  from  the  former  expression  to  the  latter  is  equivalent  to 
an  integration  considering  the  fractional  part  as  constant  during  the  in- 
tegration. This  is  vital,  and  it  is  accomplished,  physically,  by  connecting 
the  working  substance  with  a  source  possessing  constantly  the  actual  heat 
Q.  Possibly  this  is  implied  in  the  expression  "  uniformly  hot  substance  "  in 
the  law  stated  above  ;  but  if  not,  the  law  seems  to  be  defective  in  this  par- 
ticular, unless  we  resort  to  the  only  other  alternative  of  considering  the 
"  homogeneous  substance"  as  a  perfect  gas.  Heat  is  absorbed  in  doing 
work,  and  it  is  this  heat,  as  heat,  independent  of  any  particular  sub- 
stance, that  is  to  be  divided  into  equal  parts  ;  and,  having  this  concep- 
tion, it  is  apparent  that  each  part  of  the  heat  will  do  the  same  amount  of 
work.  It  is  difficult  to  determine  the  exact  meaning  of  the  expression— 
"  It  is  required  to  find  how  much  of  this  work  is  done  by  the  disappear- 
ance of  heat  ;"  for,  with  isothermal  expansion,  not  only  all  the  external 
work  is  done  by  the  disappearance  of  heat,  but  the  internal  work  also, 
according  to  which  the  words  "  of  this  "  should  be  expunged.  Another 
view,  and  probably  the  correct  one,  is,  —  it  is  required  to  find  how  much 
of  this  work  is  done  by  the  disappearance  of  an  equal  amount  of  heat. 
If  this  be  the  intended  meaning,  then  by  referring  to  Fig.  13,  p.  33,  it 
will  be  seen  that  the  area  represented  by  the  upper  strip  A  B  c  d  is  com- 
mon both  to  that  part  of  the  external  work  Vi  A  B  Vi,  and  to  the  area 
0i  A  B  fa,  representing  the  actual  heat  absorbed  ;  and,  hence,  in  per- 


392  APPENDIX  II. 

forming  this  element  of  external  work,  an  equal  amount  of  heat  will  dis- 
appear.  A  B  c  d  =  d  U  =      dp  dv;aadd(d  U)  =  dpdv  =  Abgd. 


But  d  p  is  a  direct  function  of  r,  being  limited  by  the  consecutive  iso- 
thermals  A  B  and  d  c ;  and  this  fact  is,  by  the  calculus,  indicated  in  the 
following  manner, 

\<Tr 
and  we  have 


,-r^l   dv, 
or, 


Then  the  second  law  above  declares  that  if  the  increment  of  heat  d  Q 
causes  the  work  A  b  g  d  to  be  performed  by  the  disappearance  of  an 

equal  amount  of  heat,  then  will  the  total  heat  absorbed,  <p,  A  b  n,  be  -^ 

d  Q 
times  as  much,  giving 

d(d  U) 
.         ^      dQ    ' 

The  "actual  heat  is  divided  into  equal  parts  "  by  the  successive  iso- 
thermals  of  the  substance. 

The  explanation  by  Rankine  of  this  operation  is  more  satisfactory  in 
his  original  paper  than  in  his  Steam-Engine.  (Mite.  Sc.  Papers,  p.  312.) 

His  general  law  of  the  transformation  of  energy— "  The  effect  of  the 
presence  in  a  substance  of  a  quantity  of  actual  energy,  in  causing  trans- 
formation of  energy,  is  the  sum  of  the  effects  of  all  its  parts  "  (Steam  En- 
gine, p.  309)— implies  that  the  office  of  the  working  substance  is  simply 
to  transfer  actual  energy  from  a  source  to  a  receiver  of  a  potential  form, 
as  when  the  heat  of  a  furnace  is  transferred  to  work  ;  under  which  con- 
ditions the  actual  energy  of  the  working  substance  must  be  maintained 
constant. 

Sir  William  Thomson  thus  stated  the  second  law  : 

"PROP.  II.  (Carnot  and  Clausius).—  If  an  engine  be  such  that,  wlien 
it  is  worked  backward,  the  physical  and  mechanical  agencies  in  every  part 
of  its  motions  are  all  reversed,  it  produces  as  much  mechanical  effect  as  can 
be  produced  by  any  thermodynamic  engine,  with  the  same  source  and  re- 
frigerator, from  a  given  quantity  of  heat. "  (Thomson's  Papers,  p.  178.) 

Credit  is  here  given  to  Clausius  for  a  part,  at  least,  of  the  fundamen- 
tal principle  involved,  and  hence  it  is  unnecessary  to  consider  his  views. 

We  find,  then,  that  three  of  the  principal  founders  of  the  science  of 


SECOND   LAW   OF  THERMODYNAMICS.  393 

thermodynamics — Clausius,  Rankine,  Thomson — give  as  the  SECOND  LAW 
the  principle  of  Carnot's  ideal  elementary  reversible  engine. 

Unless  the  axioms  of  these  writers,  which  are  by  Maxwell  stated  as  the 
second  law,  be  considered  as  including  the  reversible  engine,  it  appears 
to  be  improper  to  consider  them  as  the  second  law.  These  writers  stated 
them  as  axioms,  and  not  as  the  second  law. 

The  two  laws  of  thermodynamics  are  the  result  of  experience,  guided 
by  scientific  investigation.  Rankine  snys  :  "  The  laws  of  thermodynam- 
ics, as  here  stated,  are  simply  the  condensed  expression  of  the  facts  of 
experiment."  (Misc.  Se.  Papers,  p.  427 ;  Phil.  Mag.,'Oct,,  1865.) 

The  statement  of  the  second  law  referred  to  in  this  quotation  is  simply 

the  expression  T  _J?  d  v  written  out  in  words  ;  and  hence  is  equivalent 

d  T 

to  one  of  the  preceding  quotations,  and  also  to  the  one  on  page  33  of 
the  text. 

Another  statement : 

The  first  law  asserts  a  fixed,  unvarying  relation  between  heat  energy 
and  the  mechanical  energy  into  which  it  is  transmuted  ;  but  in  a  work- 
ing engine  all  the  heat  absorbed  cannot  be  transmuted  into  work,  and  the 
second  law  asserts  that  a  certain  fractional  part  of  the  heat  absorbed 
may  be  transmuted  into  mechanical  energy  when  the  substance  is  work- 
ed in  Carnot's  cycle. 


ADDENDA. 


[The  articles  in  this  Addenda  are  numbered  the  same  as  those  in  the 
body  of  the  work  to  which  the  subject-matter  pertains.] 

14.  16.  Since  air  is  not  a  perfect  gas,  the  divis- 
ions on  an  air  thermometer  will  not  be  equal  for  equal  in- 
crements of  actual  heat  absorbed  by  the  air.  The  relation 
between  p,  v,  r,  as  used  by  Thomson  and  Joule,  is  given  in 
equation  (7),  page  13.  The  experiments  of  Regnault  en- 
able one  to  determine  the  constants.  The  results  of  these 
experiments,  generalized,  enabled  Sir  William  Thomson  to 
construct  the  following  table,  in  which  the  degrees  are  for 
the  centigrade  scale  from  0  to  300,  and  d  is  the  relative 
density  of  the  air  in  the  thermometer.  If  at  0°  C.  it  be 
under  the  pressure  of  one  atmosphere,  760  mm.,  then  will 
d  =  1,  and  the  factors  of  d  will  be  the  fraction  of  one  de- 
gree on  the  centigrade  scale  by  which  the  readings  differ 
from  what  they  would  if  air  were  a  perfect  gas. 


Temperature  by 
absolttte  scale 

Temperature  centigrade 
on  air  thermometer. 

Temperature  by 
absolute  scale 

Temperature  centi- 
grade on  air  ther- 

measured from 

PT  ~  /*273-7 

measured  from 

mometer. 

0«C. 

0  =  100    -—  -^-- 

0°C. 

t  =  T  —  273.7 

P  373-7  -p-na-i 

t  =  r  —  273.7 

•  -  IOC  ;>r~  ^273-7 

/'37S-7  -^273-7 

t 

0 

t 

0 

0 

0 

160 

160—0.0970  X  d 

20 

20  +  0.0294  X  d 

180 

180—0.1363 

40 

40  +  0.0398  ' 

200 

200—0.1772 

60 

60  +  0.0361  ' 

220 

220—0.2202 

80 

80  +  0.0220  ' 

240 

240—0.2627 

100 

100 

260 

260—0.3099 

120 

120  —  00280  ' 

280 

280—0.3562 

140 

140  —  0.0607  ' 

300 

300—0.4030 

(Thomson's  Papers,  p.  100.) 


396  ADDENDA. 

22.  Numerous  equations  have  been  proposed  to  represent  the  results 
of  experiments  upon  gaseous  substances.  Rankine's  equation,  (4),  p.  13, 
is  the  most  general,  and  is  sufficiently  accurate  for  all  substances  used 
in  engineering  practice.  It  seems  a  useless  labor  to  construct  equa- 
tions that  will  represent  with  extreme  accuracy  the  experiments  made 
by  any  person,  for  the  results  of  different  experimenters  will  differ, 
and  a  formula  that  will  agree  nearly  with  one  set  will  not  agree  with 
others.  If  the  experiments  are  reliable,  like  those  of  Reguault,  the 
formula,  when  plotted,  should  exhibit  the  law  indicated  by  the  ex- 
periments, and  give  approximately  the  values  found  by  experiment. 
The  formula  pertaining  to  steam  will  be  given  in  Article  78.  The 
following  are  equations  for  carbonic  acid  gas. 

Rankine— also  Thomson  and  Joule— gave  an  equation  of  the  form 


(Phil.   Tram.,  1854,  p.  336  ;  1862,  p.  579.) 
Him  gave 

(P  +  r)  (v  —  x)  =  R  r  • 

where  x  =  "la  somme  de  volumes  des  atomes  ;" 

r  =  "  la  somme  des  action  internes." 

(Theorie  Mecanique  de  la,  Chaleur,  2"  ed.,  i.,  p.  195  ;   3'  ed.,  ii.,  p.  211.) 
Racknel,  in  1871  and  1872,  gave  the  formula 


where  a  is  a  constant  to  be  determined  by  experiment. 
J.  D.  Van  der  Waals  gave 


-     -    ' 

v  —  b         v3 

in  which  if  the  unit  of  pressure  is  one  atmosphere,  and  the  unit  of 
volume  that  which  a  kilo,  of  carbonic  acid  occupies  under  the  pressure 
of  one  atmosphere  at  the  melting  point  of  ice,  then 

R  =  0.003673, 
a  =  0.00874, 
b  =  0.0023. 

Over  de  Continuiteit  van  den  Gas  en  Vloeistoestand,   Leinden,  1873, 
p.  76,  Op.  cit.,  p.  76. 
Clausius,  in  1880,  gave 

-  P      T  c 

*     r    - 


in  which  if  the  pressure  be  in  kilogrammes  per  square  metre,  and  vol- 
ume in  cubic  metres,  we  have  per  kilogramme  of  carbonic  acid, 


ADDENDA.  397 

E  =  19.273, 
c  =  5533, 
a  =  0.000426, 
p  =  0.000494. 

This   formula  gave    results  agreeing    remarkably  well  with  those    of 
observation.     (Phil.  Mag.,  1880,  (1),  401.) 

23,  24,  25.  Thermal  lilies.  The  more  common 
thermal  lines  are  defined  in  the  body  of  the  book  ;  but  the 
following  are  sometimes  used  : 

Isopiestic,  or  Isobar  lines  are  lines  of  equal  pressure,  and, 
therefore,  on  the  plane  p  v,  are  parallel  to  the  axis  of  v. 

Isometric  lines  are  lines  of  equal  volume,  and  their  pro- 
jections on  the  plane  p  v  are  parallel  to  the  axis  of  p. 

Isengeric,  or  Isodynamic  lines  are  lines  of  equal  energy. 
In  this  case  the  internal  energy  remains  constant,  and  all 
the  heat  absorbed  during  the  change  of  state  is  transmuted 
into  external  work.  See  top  of  page  129.  If  the  gas  be 
perfect,  the  isengeric  coincides  with  an  isothermal. 

Isentropic  lines  are  lines  of  equal  entropy,  and  hence 
coincide  with  adiabatics. 

4O.  Page  33.  In  order  that  the  algebraic  expressions 
may  be  serviceable  in  numerical  problems,  the  volume  0  -y,, 
Fig.  13,  must  represent  a  definite  mass  of  the  working  sub- 
stance ;  and  we  assume  a  unit-mass  /  and  in  English  meas- 
ures let  it  be  one-  pound.  Clausius,  Zeuner,  and  others,  in 
some  cases,  include  the  internal  work  in  the  expression — 
internal  energy ;  but  we  prefer  to  apply  the  term  work  to 
all  that  part  of  the  heat  absorbed  which  is  destroyed — put 
out  of  existence  for  the  time  being — transmuted  into  an- 
other form  of  energy ;  and  if  any  part  remains,  call  it  a 
change  of  internal  energy. 

The  second  law  is  sometimes  called,  briefly,  the  revers- 
ible engine  •  or,  more  fully,  an  expression  of  the  facts  in- 
volved in  ike  simple  reversible  engine.  Isothermal  expan- 


d98  ADDENDA. 

sion  and  compression  are  the  first  fundamental  principles  of 
this  law  ;  the  second  being  the  axiom  of  Thomson  that — no 
engine  can  be  worked  with  mechanical  profit  at  a  lower 
temperature  than  that  of  the  coldest  of  surrounding  objects  ; 
and  since  absolute  cold  cannot  be  produced  in  surrounding 
objects,  it  follows  that  only  a  fractional  part  of  the  heat 
absorbed  can  be  transmuted  into  external  work. 

It  is  worthy  of  remark  that  during  isothermal  expansion 
the  heat  of  the  working  fluid  does  no  work  ;  it  is  merely  an 
agent  for  transmuting  actual  heat  energy  into  work.  If  the 
working  fluid  be  a  perfect  gas,  it  will  transfer  the  heat 
directly  from  the  source  to  the  piston  of  the  engine.  If 
the  working  fluid  be  an  imperfect  gas,  a  part  of  the  heat 
from  the  source  will  be  transferred  from  the  source  to  the 
piston  of  the  engine  and  transmuted  into  external  work, 
and  the  remaining  part  will  be  transmuted  into  internal 
work,  being  the  work  necessary  to  overcome  the  resistance 
of  the  particles  in  being  separated  during  expansion.  The 
actual,  or  kinetic,  energy  of  the  working  fluid  remains  con- 
stant during  isothermal  expansion. 

Page  34.  Sir  William  Thomson  has  proposed  two  scales 
of  absolute  temperatures.  In  the  first  scale  it  was  pro- 
posed to  consider  the  difference  of  the  temperatures  of  the 
source  and  refrigerator  as  constant  when  the  work  done  by 
a  perfect  engine  on  abstracting  a  unit  of  heat  from  the 
source  is  constant,  whatever  be  the  temperature  of  the  heat 
absorbed. 

To  get  an  idea  of  this  principle,  observe  that  in  Fig.  a, 
the  successive  divisions  represent  equal  works  done  in  suc- 
cessive elementary  engines ;  but  the  heat  absorbed,  <p^AB  q>^, 
in  doing  the  work  A  B  c  d,  is  more  than  (pl  d  c  tp^  in  doing 
the  equal  work  d  c  ij.  But  the  preceding  principle  re- 
quires that  the  heat  absorbed  along  y  z  must  equal  that 
along  A  B,  while  the  elementary  works  done  in  the  cycles 
must  be  the  same  ;  therefore,  to  represent  this  case  y  z  must 


ADDENDA. 


399 


be  prolonged  so  that  the  area  under  it  and  between  the  two 
adiabatics  through  its  extremities  shall  equal  cp^AB  (f>^  and 
the  elementary  strip  under  y  z  must  be 
so  narrow  that  its  area  shall  equal  A  B  c  d. 
To  find  the  symbolic  expression,  let 

H  =  (p^  A  B  (p3  =  the  heat  ab- 
sorbed, in  foot-pounds, 

dII=ABcd  =  the  heat  trans- 
muted into  work, 

d  t  =    the   difference   of    absolute 
temperatures  between   the  source   and 
refrigerator, 

—  =  the  fractional  part  of  the  heat  absorbed  that  is 
transmuted  into  work  per  unit  of  tempera- 
ture ; 


,  V  Va  v» 
FIG.  a. 


then 


the  work  =   t  If  d  t  —  d 


d  II 


But  according  to  the  principle  above  stated,  p  is  not  only 
independent  of  the  temperature,  but  the  ratio  of  the  left 
member  is  to  be  constant  ;*  hence  jn  is  constant,  and  we 
have  by  integrating  between  the  limits  2Il  and  Ht  for  heats, 
and  tl  and  tt  for  temperatures, 


— that  is,  if  the  differences  on  the  scale  of  the  thermometer 


*  Or  integrating,  we  have  : 


Work  =  H,  —^  =  ^(1  —  6  ). 

But,  in  this  scale,  the  work  is  a  function  of  the  difference  of  temperatures, 
whatever  be  the  temperature  of  Hi,  and  this  condition  requires  that  /* 
should  be  constant.  This  relation  is  clearly  shown  in  equation  (a). 


4<X)  ADDENDA. 

increase   arithmetically,  the   ratio   of  heats   of  the  source 
and    refrigerator    will    increase    geometrically. 
The  efficiency  would  be 


and  the  work  done 

H.-H^H^l-e-^'-1*}-,  (a) 

or,  the  work  will  be  the  same  for  each  degree  on  the  scale 
for  the  same  amount  of  heat  absorbed,  regardless  of  its  tem- 
perature. 

Let  this  scale  have  180  divisions  between  the  melting 
point  of  ice  and  the  boiling  point  of  water  ;  then,  since  the 
efficiency  of  the  perfect  elementary  engine  worked  between 
these  temperatures  is 

//.  -  ff,  _  _  180  180 

H,        ~  460.66  +  212  "  672.66  ~ 
-iso*  _      180 

~  672.66  ; 
.-.//  =  0.00173. 

According  to  the  absolute  scale  in  ordinary  use, 


Hv  T,  T,  +  460.66 

the  last  fraction  being  applicable  to  the  Fahrenheit  scale. 
Let  this  scale  and  the  former  absolute  scale  above  considered 
both  be  numbered  212  at  the  boiling  point  of  water,  then 

T,  =  212  =  ta 

and  dropping  the  subscript  2  we  have  for  the  relation 
between  t  and  T, 

212  -  T  _     _         I 
672.66    =  ^212-0' 

from  which  we  find 


ADDENDA.  401 

672.66 


2.30258  log,. 


H  =  0.00173 

by  means  of  which,  if  any  degree  T  be  given  o,n  the 
Fahrenheit  scale,  the  corresponding  number  t  may  be  found 
on  Thomson's  first  absolute  scale.  If  T  =  —  460.66, 
t  =  —  oo  ;  that  is,  the  absolute  zero  of  this  scale  is  minus 
infinity.  The  higher  the  temperature  T,  more  and  more 
degrees  will  be  required  to  make  a  degree  on  the  absolute 
scale. 

This  scale  is  not  practical.  It  was  devised  while  Thom- 
son held  to  the  old  theory  of  heat  (or  caloric),  which 
maintained  that  heat  was  material,  and  hence  could  not  be 
converted  into  work.  He  said  :  "  The  conversion  of  heat  (or 
caloric)  into  mechanical  effect  is  probably  impossible,  cer- 
tainly undiscovered."  (Phil.  Mag.,  1848,  (2),  315.)  Work 
was  then  supposed  to  be  derived  from  heat  by  letting  it 
down  from  a  hot  body  to  a  cold  one  without  diminishing 
the  quantity  of  heat  ;  just  as  work  is  obtained  from  water 
by  letting  it  down  from  one  level  to  a  lower  one  while 
passing  through  a  motor  without  diminishing  the  quantity 
of  water. 

(Phil.  Mag.,  1848,  (2),  313;  Trans.  R.  S.  E.,  XX. 
(1851),  273  ;  Phil.  Mag.,  1852,  (2),  106  ;  Thomson's  Papers, 
Vol.  I.,  p.  139.) 

Making   /*  =  —  gives,  by  integration, 


or  Thomson's  second  scale.  (Trans.  It.  S.  E.,  1854,  p.  125; 
Phil.  Mag.,  1856,  (1),  216.)  A  comparison  of  this  scale 
with  the  air  thermometer  is  given  in  Article  14  of  this 
Addenda. 


402  ADDENDA. 

Page  38.  In  the  third  line  from  the  top  it  will  be  ob- 
served that  the  argument  depends  upon  an  inference, 
"It  is  inferred"  &c.  No  absolute  proof  of  equation  (21) 
has  been  made.  Rankine  deduced  it  in  a  short  way,  found- 
ing it  upon  the  hypothesis  that — in  a  given  mass  of  a 
substance  the  quantities  of  sensible  heat  are  proportional  to 
their  absolute  temperatures.  (Misc.  Sc.  Papers,  pp.  50, 
56,  376,  377,  409;  Phil.  Trans.,  1854.)  Thomson  and 
Joule  established  it  by  a  long  and  very  delicate  series  of 

experiments,  determining  that  //=:-,  establishing  this  value 

within  f$  ff  of  its  actual  valne.  Clausius,  in  his  later  work, 
established  it  by  a  process  of  analysis,  but  it  is  somewhat 
obscured  by  the  more  general  equations  in  which  it  is  in- 
volved. Yet  there  is  no  question  as  to  the  correctness  of 
this  equation ;  for  not  only  do  Thomson  and  Joule's  experi- 
ments prove  it  to  be  as  nearly  mathematically  exact  as  it  is 
possible  by  means  of  physical  experiments,  but  it  produces 
substantially  correct  results  when  applied  to  various  prob- 
lems in  this  science  ;  and  this  is  the  best  test  of  a  physical 
law.  The  Newtonian  law  of  universal  gravitation  is  accepted 
as  mathematically  exact  for  all  problems  to  which  it  is 
applied  involving  finite  distances,  although  the  law  does  not 
admit  of  an  absolute  proof. 

The  hypothesis  of  Rankine,  referred  to  above,  was  criti- 
cised by  Clausius,  and  Rankine  modified  it  thus :  "  A 
change  of  real  specific  heat,  sometimes  considerable,  often 
accompanies  the  change  between  any  two  of  those  condi- 
tions" of  solid,  liquid,  or  gaseous.  This  definition  was  not 
accepted  by  Clausius  as  correct.  (Clausius  On  Heat,  1879, 
pp.  345-348 ;  Rankine's  Prime  Movers,  p.  307  ;  Phil. 
Mag.,  Ser.  4,  Vol.  VII.,  p.  10 ;  Pogg.  Ann.,  Vol.  CXX., 
p.  426  ;  Phil.  Mag.,  Ser.  4,  Vol.  XXX.,  p.  410.)  If  this 
principle  be  not  rigorously  and  universally  exact,  it  is,  so 
far  as  known,  correct  within  the  limits  of  error  of  observa- 


ADDENDA.  403 

tion  for  imperfect  gases;  and  these  are  the  substances  to 
which  this  science  is  very  largely  applied. 

42.  Page  43.  During  isothermal  expansion  at  a  given 
temperature,  f,  a  certain  amount  of  external  work  will  be 
done,  as  vl  d  e  v^  Fig.  #,  and  a  certain  amount  of  internal 
work  which  we  assume  is  not  yet  known.  If  the  same 
amount  of  expansion  be  performed  at  a  temperature  r  -\-  d  T, 
the  external  work  v1  A  B  vt  will  be  done,  and  a  certain 
amount  of  internal  work,  which,  as  before,  may  be  un- 
known. One  of  the  brilliant  points  in  Kankine's  establish- 
ment of  equation  (21)  was  his  conception  that  the  differ- 
ence between  the  external  works  for  two  equal  isothermal 
expansions  in  which  the  temperatures  of  the  source  differed 
innnitesimally,  being  r  in  one  case  and  r  -\-dr  in  the  other, 
equalled  the  work  done  by  an  elementary  engine  for  the 
same  isothermal  expansion,  the  temperature  of  the  source 
being  that  of  the  higher  temperature,  T  -}-  d  r,  and  of  re- 
frigerator that  of  the  lower,  or  r.  In  other  words,  the  in- 
crement of  external  work  due  to  an  elementary  increase  of 
the  temperature  of  the  source  equalled  the  increased  incre- 
ment of  the  heat  absorbed.  Or  the  increment  A  B  c  d  of  the 
heat  (pl  1)  c  <7>2  equals  the  increment  A  B  c  d  of  the  external 
work  vl  A  B  v,.  This  is  equivalent  to  asserting  that  the 
difference  of  the  internal  works  done  during  the  perform- 
ance of  the  external  works  vl  d  c  v^  and  vl  A  B  v^  is  zero  ; 
or,  at  most,  a  difference  of  the  second  order  compared  with 
the  difference  of  external  works. 

As  a  verification  of  this  principle  for  a  particular  case,  let 

the  equation  of  the  gas  be  p  =  -R  --  ^r~  i  1  tnen  w^  tne 
area  of  one  of  any  one  of  the  strips  in  99,  A  B  <py  be 


dcij  = 

as  already  given  in  Exercise  2,  page  44.    The  external  work 
for  isothermal  expansion  will  be 


404  ADDENDA. 


as  in  Exercise  4,  page  45.     The  differential  of  the  latter  in 
regard  to  r  gives 


which  is  the  same  as  found  above  for  d  c  ij. 

43.  In  the  Exercises  on  page  44,  it  must  be  under- 
stood that  they  refer  to  a  unit-mats,  and  when  numbers 
are  given,  they  refer  to  one  pound  of  the  gas. 

48.  By  consulting  the  records  of  Regnault's  experiments, 
one  becomes  impressed  with  the  large  amount  of  work  done 
by  him  and  the  extreme  accuracy  with  which  his  experi 
ments  were  made. 

51.  By  a  purely  analytical  relation  between  the  physical 
properties  of  a  perfect  gas,  it  has  been  shown  that 

y  -  1  +  ^3  =  1.405285. 

(Phil.  Mag.,  1885,  (1),  520.) 

58.  Page  63.  Air  has  been  compressed  to  500  pounds  per 
square  inch  for  use  in  the  Yincennes-  Villa  tramway,  (La 
Nature  Sc.  Am.  Sup.,  1888,  Mar.  17,  p.  10167.)  Air  has 
been  compressed  to  1000  pounds  per  square  inch  for  use  in 
dynamite  guns. 

58.  Exercise  15,  page  71. 

II  =  a  7>n. 

Equations  (B)  give 

vdp-\-ypdv  =  an(Y—  1)  V-1  ef  0. 
Let 

v  =  x,   p-y,   P=-,  Q  =  a  n  (y  -  1)  x  "-»  =  B  x  n-» 


ADDENDA.  405 

and  the  equation  becomes 


where  P  and  Q  are  functions  of  x.     This  is  a  differential  equation  of 
the  first  order  and  first  degree. 

To  find  the  integral,  first  let  Q  =  0,  then 


-  f 

j 


Pdx 


where  G.  is  a  function  of  x  instead  of  a  constant  of  integration.     Differ- 

entiating, 

-fet* 


Y-=-PGxe 
dx 


-fPdx  /Pdx 

.'.Qdx=e  .dC,;    .'.  Cx=qe  dx; 


—fPdx  f 


Pdx 


••P  =  ^TT-y" 
Exercise  16  may  also  be  reduced  to  the  linear  form,  giving 


Exercise  17  gives  the  differential  equation 

dp^  _  na(y  —  l)  va~l  -  yp 
dv         nb(\  —  y)pu~1  —  v 


71.  Page  89.  The  melting  point — or  freezing  point — of 
liquid  carbon  disulphide     —  116°  C. 

"      absolute  alcohol          -  130.5°  C. 
Alcohol  becomes  viscid  at  —  129°  C. 

(Phil.  Mag.,  1884,  (1),  490.) 

12.  Page  90.  By  experiment  it  has  been  found  that  the 
melting  point  of  ice  is  raised  0.0066°  C.  by  a  reduction  of 
pressure  from  760  mm.  to  5  mm.  (Phil.  Mag.,  1887,  (2), 
295.) 


406  ADDENDA. 

74.  Of  liquids  and  saturated  vapor.  Regnault 
found  the  latent,  heat  of  evaporation  by  determining  the 
total  heat  and  the  heat  of  the  liquids  independently,  and 
taking  their  difference. 

Total  heat  of  liquids,  being  the  number  of  ther- 
mal units  necessary  to  raise  the'  temperature  of  a  unit-mass 
from  that  of  melting  ice  to  t  degrees  centigrade,  as  deter- 
mined by  Regnault,  at  atmospheric  pressure. 

Substance.  Number  of  Thermal  Units. 

Water  ..............  q  =  <  +  0.00002  <8  +  0.0000003  t3. 

Alcohol  ............  q  =  0.54754  1  +0.001122  1*  +  0.000002  13. 

Ether  ..............  q  =  0.52901  1  +  0.0002959  <*. 

Chloroform  .........  q  =  0.23235  t  +  0.0000507  «*.  }•  (1) 

Chloride  of  carbon  .  .q  .=  0.19788  1    +0.  0000906  1*. 
Acetic  acid  ..........  q  =  0.506403  1  +  0.000397  ?-. 

Bisulphide  of  carbon  q  =  0.  23523  1  +  0.000088  <*. 
The  general  law  of  these  equations  may  be  represented  by  the  empir- 
ical equation 

q  =  a,  t  +  6,  P  +  <r,  t\  (2) 

To  reduce  these  to  English  units,  we  observe  that  q,  the 
number  of  thermal  units,  is  independent  of  the  unit  of 
mass,  or  weight,  since  the  ratio  between  the  quantities  of 
heat,  in  this  case,  and  the  respective  quantities  of  liquid  ex- 
perimented upon  will  be  constant,  but  will  be  dependent 
upon  the  thermometric  scale. 

Since  the  degree  in  the  British  thermal  unit  is  -|  that  in 
the  French,  the  number  of  British  thermal  units  will  be 
£  times  the  number  of  French  units  for  the  same  temper- 
ature ;  and  if  T  be  the  temperature  on  the  Fahrenheit 
scale  we  have 


hence,  for  water  we  would  have,  q  denoting  the  n  uirJ><  r  of 
B.  T.  TL, 

q  =  g  q  =  |  [|  (T  -32)  +  0.00002  [§  (T  -  32)]«  +  0.0000003  [$  (T-  32)]3]; 
and  similarly  for  the  other  liquids.  Substituting  and  re- 
ducing, we  have 


ADDENDA.  407 

Substance.  No.  of  B.  T.  U.  at  temp.  T. 

"Water,  q  = 

—  31.991656  +  0.99957333  T+  0.000002222  T*  +  0.0000000926  I 
Alcohol,  g  = 

-  16.903214  +  0.509543  T     +  0.00056407  T1 

+  0.00000061 7284  T*. 
Ether,  q  = 

-  16.759986  +  0.518489  T     +  0.00016439  T3. 
Chloroform,  q  — 

-  7.406358  +  0.230547  7      +  0.00002817  T. 
Chloride  of  carbon,  q  = 

-  6.280619  +  0.194659  T     +  0.00005033  712. 
Acetic  acid,  q  — 

-  15.979005  +  0  492287  T     +  0.00022055  T*. 
Bisulphide  of  carbon,  q  = 

-  7.480711  +  0.232314  1     +  0.00004555  T. 
The  general  law  will  be 

The  figures  in  equations  (3)  were  obtained  by  carrying 
out  the  decimals  to  many  more  places,  and  then  retaining 
the  above  to  the  nearest  unit  for  the  right-hand  figure. 

The  specific  heat  at  any  temperature  will  be  the 
differential  coefficients  of  the  preceding  expressions,  which, 
for  water,  will  be 

^-2  =  1  +  0.00004  t  +  0.0000009  f,  (5) 

per  degree  centigrade,  and 

",=  0.999573  +  0.000004444  T -\-  0.00000027768  T,  (6) 


per  degree  Fahrenheit. 

These  results  for  water  are  not  exactly  the  same  as  Ran- 
kine's  or  Bosscha's,  given  in  Article  95,  but  any  one  of  them 
is  sufficiently  exact  for  ordinary  practice. 

74,  85.  Total  heat  of  vapor.  This  expression  in- 
cludes the  heat  imparted  to  the  liquid  in  raising  its  tem- 
perature from  that  of  the  melting  point  of  ice  to  that  at 
which  the  vapor  is  generated  added  to  the  heat  necessary  to 
evaporate  the  liquid  at  the  higher  temperature.  The  latent 


408  ADDENDA. 

heat  of  evaporation  includes  both  internal  and  external 
work ;  the  external  being  the  work  of  enlarging  the  volume 
at  the  pressure  corresponding  to  the  higher  temperature,  and 
may  be  represented  by  the  work  done  by  a  pound  of  satu- 
rated steam  in  pushing  a  piston  against  a  constant  resistance 
up  to  the  point  of  cut-off  in  an  engine,  and  the  internal, 
that  of  overcoming  the  mutual  attractions  between  the 
molecules.  Let  h  be  the  number  of  heat  units  necessary  to 
raise  one  kilogram  of  a  liquid  from  0°  C.  to  a  temperature  t, 
and  vaporize  it  at  that  temperature  ;  then  Regnault's  ex- 
periments may  be  represented  by  the  following  empirical 
formulae  : 

Number  of  heat  unite  in  the  "  total  heat 
of  vapor  "  in  French  thermal  unite. 

Water h  =  606.50  +  0.305  t. 

Ether h  =    94.00  +  0.45000  t  -  0.00055556  P. 

Acetic  acid h  =  140.50  +  0.36644  t  -  0.000516  P. 

Chloroform h  =    67.00  +  0. 1375  t. 

Chloride  of  carbon  . .  h  =    52.00  -j-  0.14625  t  -  0.000172  P. 
Bisulphide  of  carbon,  h  =    90.00  +  0.14601  t  -  0.0004123  t*.     J 
For  English  units,  if  h  be  the  number  of  heat  units  in 
one  pound  of  the  substance  on  the  Fahr.  scale,  then  for 
water  we  would  have 

A  =  f  h  =  f  [606.5  +  0.305  X  \  (T  —  32)] 
=  1091.7     4-  °-3°5  (T  —  32) 
=  1081.94  4-  0.305  T-, 
.  •  .  H  =  841829    4-  237.29  T, 

which  differs  slightly  from  Eq.  (95),  page  111,  because  some 
fractions  were  omitted  in  determining  the  latter.  In  this 
manner  we  find  the  following  results : 

Number  of  B.  T.  U.  in  the  "  total  heat  of  vapor  " 
of  1  lb.,  Fahr.  scale,  above 0*  F. 

Water h  =  1081.94  +  0.305  T. 

Ether h  =  154.4839481  +  0  46974324  T-  0.000308633  T>. 

Acetic  acid. .  h  =  240.880363    +  0.3847866  T  -  0.000286666  T2.    I  (?) 
Chloroform.,  h  =  116.2  4-  0.1375  T. 

Ch.  of  carbon  h  =  88.82215  +  0.1523655  T  -  0.00009555  T*. 
B.  of  carbon,  h  =  157.093127  +  0. 16066955  T  -  0.000229055 T.        J 
General  equation 

A  =  o,  +  6,  T-c*  T'.  (8) 


ADDENDA.  409 

74.  Latent  heat  of  evaporation.  Subtracting  the  "  heat 
of  the  liquid "  from  the  "  total  heat  of  the  vapor  "  gives 
the  latent  heat  of  evaporation ;  hence 

^e  =  A  -  q ; 

and   making  the    substitutions   from   above,  we   have   the 
following  results : 

Latent  heat  of  evaporation,  being  the  No.  of  French 
Substance.  heat  units  in  one  kilo,  of  the  vapor 

at  the  boiling  point. 

Water he  =  606.5  -  0.695  t     -  0.00002 1* -  0.0000003 1* 

Ether he  =    94.0  -i  0.07901  t  -  0.0008514  P. 

Acetic  acid he  =  140.0  -  0.13999  t  -  0.0009125  P.  . 

Chloroform he  =   67.0  -  0.09485  t  -  0.0000507  f. 

Chlo.  of  carbon. .  h.  =    52.0  -  0.05173  t  -  0.0002526  <2. 
Bisulp.  of  carbon  he  =    90.0  -  0.08922  t  -  0.0004938  P. 
In  English  units  these  become : 

Latent  heat  of  evaporation,  being  the  heat  necessary 
Substance.  to  evaporate  one  pound  of  the  substance  at  the 

boiling  point,  in  B.  T.  U. 

Water A.  =  1121.7  -  0.6946  T  -  0.000002222  7"  - 

0.0000000926  T3. 

Ether A.  =  171.24  -  0.0487  T  -  0.000473  T\ 

Acetic  acid A.  =  256.86  -  0.1075  T  -  0.000507  T\ 

Chloroform Ae  =  123.60  -  0.0930  T  -  0.000282  T*. 

Chlo.  carbon A.  =  95.103  -  0.0423  T  -  0.0001403  T*. 

Bisulph  carbon.  .A.  =  164.57  -  0.0716  T  -  0.0002746  7"s. 
Alcohol    h,  =  527.04  -  0.92211  T  -  0.000679  I  \ 

General  equation  : 

Jit  —  «4  _  J4  T  —  c4  T*  —  dt  T3.  (11) 

These  for  English  units  and  absolute  temperature  on  the 
Fahrenheit  scale  become  : 

Latent  heat  of  evaporation  in  B.  T.  U. ,  absolute 
Substance.  temperature,  r. 

Water    A.  =  1442.474  -  0.751472  T  +  0.0012538  r2  -  ] 

0.0000000926  r\ 

Ether A.  =    93.3214  +  0.3870  T  -  0.000473  r2. 

Acetic  acid A.  =  197.925  +  0.3595  r  -  0.0005070  r5.  \ (12) 

Chloroform A,  =  160.4924  -  0.0571  r  -  0.0000282  f\ 

Chlo.  of  carbon. .  A,  =    85.0245  +  0  0865  r  -  0.0001403  r*. 

Bisulp.  carbon. . .  A.  =  140.1806  +  0.1810  r  -  0.0002743  t>.          J 

General  equation  : 

fee  —  a&  _j_  J5  r  +  c6  r2  -+-  rf»  r3.  (13) 


410  ADDENDA. 

The  effect  of  retaining  the  smaller  decimals  will  be  ap- 
parent by  comparing  the  above  results  for  water  with  the 
corresponding  ones  on  page  95.  Those  on  the  latter  page 
are  considered  sufficiently  accurate  for  practice.  None  of 
them  can  be  relied  upon  for  temperatures  much  outside  of 
those  in  the  experiments  upon  which  they  are  founded. 
The  two  equations  are  not  very  different  within  the  range 
of  temperatures  ordinarily  used  in  practice. 

To  find  the  heat  which  does  the  disgregation  worl\  we 
must  find  the  external  work  done  during  evaporation.  This 
may  be  done  as  follows  :  The  pressure  for  the  absolute  tem- 
perature of  the  vapor  is,  equation  (80),  page  97, 

B      C 
logp  =  A  ----?• 

"We  have  computed  the  following  constants  by  means  of 
Regnault's  experiments.  They  are  for  degrees  Fahrenheit 
and  pounds  per  square  foot : 

Fluid.  A.  log  B.  log  C. 

Steam 8.28203  3.44L474  5.583973 

Ether 7.5641  3.3134249  5.2173549 

Alcohol 8.6817  3.4721707  5.4354440 

Bisulph.  Carbon..7.4263  3.3274293  5.134414(5 

Chloroform 4.3807     B  is  3.288394  negative.  6.1899631 

Snip.  Dioxide... 7.3914  3.1580608  5.3667327 

Naphtha 6.4618  2.949092  5.796469 

Ammonia 8.4079  3.34154 

Mercury 7.9711  3.74293 

Far  steam.  For  ether. 

B  =  2763.59,  B  =  2057. 3, 

C  =  383683.  C  =  164950. 

Now  find  the  volume  of  a  pound  of  the  vapor  by  means 
of  equation  (84),  or  the  approximate  one,  (86).  Thus, 


ADDENDA.  411 


V         rdp     '  ^  dp. 

Tr  Trr 

The  value  of  he  is  given  by  equations  (10)  and  (11),  and 
substituting,  gives 

a  -  I  T  -  c  T* 

«/. 


1  +  ^)2, 


23026 


This  is  the  outer  work.     The  disgregation  work  will  be 
p  =  II,  -p  (v,-  v,}.  (11) 

Zeuner,  by  this  laborious  process,  computed  the  disgre- 
gation work  for  a  range  of  temperatures,  and  for  various 
substances,  and  assumed,  arbitrarily,  that  they  followed  the 
law 

p  =  at  -  I,  t  —c,f, 

and  determined  the  constants    by  means  of   his    previous 
computations,  and  obtained  the  following  results  : 
For  French  units. 

Ether  ..........  p=    86.54  -  0.10648  1  -  0.0007160  f. 

Acetic  acid  .....  p  =  131.63  —  0.20184  £  —  0.0006280  t. 

Chloroform  ____  p=    62.44  —  0.11282  1  —  0.0000140  f. 

Ohio,  of  carbon  p  =    48.57  -  0.06844*5  -  0.0002080  f. 

Bisulp.  of  carbon  p  =    82.79  —  0.11446  1  -  0.0004020  f. 

For  saturated  steam,  the  outer  work  may  be  found  verv 
nearly  from  a  table  of  the  properties  of  saturated  steam, 
by  multiplying  together  the  corresponding  pressures  and 
volumes.  The  product  will  be  p  u,,  Eq.  (14).  The  temper- 
atures being  given  in  such  a  table,  we  may  find  the  total 
latent  heat  of  evaporation  by  means  of  equation  (78), 
page  95. 

Eeduce  the  pressures  to  pounds  per  square  foot,  if  neces- 
sary; and  substitute  in  equation  (14),  to  find  the  disgre- 
gation work. 


412  ADDENDA. 

The  latent  heat  of  evaporation,  as  commonly  used,  might 
be  called  the  APPARENT  latent  heat  of  evaporation  /  and  the 
disgregation  work,  the  REAL  latent  heat  of  evaporation. 

Page  96.  The  latent  heat  of  evaporation  is  reported 
differently  by  different  authors.  Thus  I  find  that  one  author 
gives  for  oil  of  turpentine,  123,  another,  133,  and  still  an- 
other, 184;  and  I'have  not  ascertained  which  is  correct. 
For  ether,  164.0,  162.8.  Alcohol,  304.8,  372.7,  385.  Naph- 
tha, boiling  point,  306,  141°  F.  ;  latent  heat  of  evaporation, 
184,  236. 

Densities  of  some  vapors  compared  with  that  of  air  when 
near  their  boiling  points  : 

Atmospheric  air 1.000 

Steam 0.6235 

Alcohol  vapor 1.6138 

Sulphuric  ether  vapor 2.5860 

Vapor  of  oil  of  turpentine 3.0130 

Vapor  of  mercury 6.976 

The  densities  of  vapors  at  the  boiling  points  of  the  liquids 
are  approximately  inversely  as  their  latent  heats  of  evapo- 
ration. 
Thus, 

Density  of  vapor  of  alcohol  =  1.6138  _  .    _ 

Density  of  steam  =  0.6235  ~ 

Latent  heat  of  evaporation  of  steam      =    966.1  _ 
Latent  heat  of  evaporation  of  alcohol  =    372.7  ~ 

76.  Eankine,  in  his  article  On  the  Centrifugal  Theory 
of  Gases,*  deduces  an  equation  of  the  form, 

logp  —  a  -  -, 

for  the  relation  between  the  pressure  p  and  absolute  tem- 
perature r  of  saturated  vapor.     It  was  found,  according  to 

*  Mis.  Sc.  Papers,  p.  43 ;  Phtt.  Mag.,  Dec  ,  1851. 


ADDENDA.  413 

Regnault's  experiments  and  others,  to  be  accurate  for  a 
limited  range  of  temperatures  only.  Kankine  then  proceeded 
to  find  an  empirical  formula  that  would  represent  more 
accurately  a  greater  range  of  temperatures,  and  was  led,  by 
analogy,  to  try  a  third  term  containing  the  inverse  square 
of  T.  thus  giving 

B       (T 
logp  =  A  -  —  -  -,. 

which  was  found  to  represent,  quite  satisfactorily,  the  re- 
sults of  experiments  upon  steam,  mercury,  alcohol,  ether, 
turpentine,  and  petroleum. 

Some  fifty,  or  more,  formulas  have  been  devised  to  ex- 
press the  relation  between  the  pressure  and  temperature  of 
saturated  steam ;  all  of  which  are  sufficiently  accurate  for 
certain  small  ranges  of  temperature  and  pressure.     Ran- 
kine's,  given  above,  is  the  most  accurate  for  a  large  range. 
Some  of  the  most  celebrated  of  the  other  formulas  are  : 
Dulong  and  Arago's,  for  pressures  above  four  atmospheres 
p  -  (0.4873  +  0.012244  <)5  Ibs.  per  sq.  in., 

t  being  the  temperature  centigrade. 
Mallet's,  from  1  to  4  atmospheres, 


P=(TiTiSF      Ibs.  persq.  in.  (15) 


Tredgold's  is  the  same,  except  that  175  is  substituted  for  111.78. 
Pambour's,  from  1  to  4  atmospheres,  t  °  C. 


Roche's 


p  =  a^  (17) 

Regnault  made  three  equations  ;  the  first  applying  from  —  30°  C.  to 

0°  C. 

log  p  =  a  +  b  a",  millimetres,  (18) 

in  which  a  =  -  0.08038  ;    log  b  =  9.6024724  -  10  ;   log  a  = 
n  =  32°  +  t. 


414  ADDENDA. 

From  0°  to  100°  C. 

l-jgp  =  a  -  b  a1  +  c  p,  millimetres,  (19) 

in  which  a  =  4.7384380  ;    log  b  =  0:6116485  ;    log  c  =  8.1340339  -  10  ; 
log  a  =  9.9967449  -  10  ;  log  ft  =  0.0068650. 
From  100°  to  220°  C. 

log  p  =  a  -  b  a"  +  c  pa,  (20) 

in  which  a  =  5.4583895  ;  log  b  =  0.4121470  ;  log  c  =  7.7448901  -  10  ; 
log  a-  9.99741212  -  10  ;  l<g  ft  =  0.007590697. 

Zeuner,  by  a  special  investigation,  deduced  a  formula  applicable  both 
to  saturated  and  superheated  steam,  which  is 

pv  =  RT  -  Cp*,  (21) 

in  which  R  =  P^t  although  it  is  better  to  consider  R,  C,  n,  as  con- 
stants to  be  found  from  experiment.  For  steam  this  becomes 

p  v  =  50.933  r  -  192.50  p  X,  (22) 

in  which  p  is  the  pressure  in  kilograms  per  square  metre,  and  v  the  spe- 
cific volume  in  kilograms  per  cubic  metre.  This  formula  is  suffi- 
ciently accurate  from  0.2  of  an  atmosphere  to  15  atmospheres  and,  so  far 
as  tested,  gives  good  results  for  superheated  steam. 

Hr.  Ritter,  from  a  discussion  of  experiments  by  Him,  proposed  the 
equation* 


in  which 

R  =  4.653,  5  =  1043800  ; 

which  gives  values  for  v  agreeing  almost  exactly  with  the  results  found 
by  equation  (84),  page  98.     But  it  is  too  complex  for  analytical  discus- 
sions. 
Unwin  proposed  the  equation  f 

t°ffi*P  =  a  ~  ~  (24) 

=  7.5030-^ 

for  saturated  steam  ;  in  which  p  is  in  millimetres  of  mercury  and 
TO  =  —  273,  on  the  centigrade  scale.  This  equation  is  nearly,  but  not 
quite,  as  accurate  as  Rankine's  ;  and  possesses  some  advantages  for  an- 
alytical purposes.  Unwin  finds,  for  the  latent  heat  of  evaporation, 

*  Pogg.  Ann.  (2),  iii.  (1878),  447. 

t  On  the  relations  of  temperature,  pressure  and  volume  of  saturated 
steam.—  Phil.  Mag.,  1886,  (1),  299-308. 


ADDENDA.  415 

h*  =  ™g-(i.m-tgloPr  <25> 

and  for  the  specific  volume  of  saturated  steam, 
10821  hK 

«•  =  *  +  ?(«-%..„)• '  (36) 

Also,  for  alcohol ; 

logp  =  7.448  -  ^-9-  (27) 

For  ether ; 

log  p  =  6.9968  -  ^~.  (28) 

For  carbonic  acid ; 

=  8.4625  -  ?^H  (29) 


The  author,  considering  steam  as  an  imperfect  fluid—  or 
gas  —  both  when  saturated  and  superheated,  applied  Ran- 
kine's  general  equation  for  imperfect  gases  with  the  follow- 
ing results  :  — 

The  general  equation  being 

pV  =  £T-a0-^-%-  &c,  (30) 

and  first  assuming  that  the  resulting  equation  might,  pos- 
sibly, be  of  the  same  form  as  his  for  carbonic  acid  gas,  I 

made  a0  =  0,  al  =  —  ,  and  all  the  remaining  terms  zero,  giv- 


pv  =  Er-.  (31) 

But  this  gave  no  satisfactory  result.     I  then  assumed 

I 
>?•>> 

and  all  succeeding  terms  zero,  as  Rankine  considered  that 
a0,  aa  &c.,  were  inverse  functions  of  0,  thus  giving 

p  v  =  E  r  -  ~  (32) 


416  ADDENDA. 

To  find  the  constants  R,  5,  n,  requires  three  contempo- 
raneous values  of  the  variables  p,  v,  T.  I  determined  values 
of  y  by  assuming  several  values  for  p  and  T  by  equation 
(85),  page  98,  and  compared  the  results  with  the  recent 
tables  of  Professor  Peabody,  and  in  no  case  did  they  differ 
by  more  than  .02  and  in  most  cases  they  agreed  exactly  to 
the  second  decimal  figure — and  this  too  notwithstanding  he 
used  Regnault's  equations  for  the  relation  between  pres- 
sures and  temperatures,  while  I  used  Rankine's,  being  equa- 
tion (80),  page  97 ;  but  using  for  constants  my  own  values 
on  page  98,  instead  of  those  computed  by  Rankine.  So 
close  an  agreement  was  not  anticipated  under  these  condi- 
tions ;  and  where  the  difference  was  greatest  it  might  pos- 
sibly have  been  less  had  I  used  more  decimals.  I  there- 
fore use  Peabody's  tables  with  confidence.  I  have,  how- 
ever, used  v,  =  26.58  for  the  specific  volume  of  saturated 
steam  at  212°,  and  pressure* of  14.7  Ibs.  per  square  inch  as 
computed  on  page  102  ;  but  all  other  values  I  have  taken 
from  the  tables. 

Using  as  arguments  the  three  sets  of  values  : 

Pl  =  14.7  X  144,      p,  =  100  X  144,      p,  =  160  X  144, 

<y,  =  26.58,  v,  =  4.38,  v9  =  2.83, 

r,  =  672.66;  r,  =  788.16;  r,  =  824.06; 

Ifind 

R  =  96.95,        b  =  18473,        n  =  0.22 ; 

giving  the  equation 

p  v  =  96.95  r  -  ^j£?  (33) 

More  decimals  in  the  value  of  n  would  make  vn  slightly 
larger,  and  as  only  two  are  retained,  a  comparative  compen- 
sation would  be  produced  by  increasing  the  value  of  the 
numerator,  and  the  result  of  a  few  trials  justified  this  modi- 
fication, and  we  finally  have 


ADDENDA. 


417 


For  saturated  and  superheated  steam 
«,  =  96.95  r  - .«™ 


(34) 


in  which  pt  will  be  in  pounds  per  square  foot,  and  v  the 
cubic  feet  per  pound.  If  p  be  in  pounds  per  square  inch 
and  v  the  cubic  feet  per  pound ;  then 


=  0.6732 


(35) 


In  order  to  test  this  equation,  I  made  the  following  table 
for  saturated  steam ;  taking  from  the  table  the  values  of  v 
and  T,  and  comparing  the  computed  and  tabular  values 
of  p. 


TEST  OF  EQUATION  (34). 

SPECIFIC 
VOLUME 

CU.  FEET. 
V. 

TEMPERATURE. 

PRESSURE. 

Fraction 
of  error. 
Columns 
5  and  6. 

DegreesF. 
T. 

Absolute. 

Pounds 
per  sq.  in. 
P  . 

Pounds  per  sq.  foot. 

Calculated. 
Eq.  (34). 
Pt 

Tabular 
value. 

90.31 
53.37 
26.58 
7.10 
5.42 
4.40 
2.83 
2.294 

153.1 
176.9 
212.0 
292.5 
311.8 
327.6 
363.4 
381.7 

613.76 
617.56 
672.66 
753.16 
772.46 
778.26 
824.06 
842.36 

4 
7 
14.7 
60 
80 
100 
160 
200 

583 
1013 
2116 

8597 
11462 
14550 
23030 

28882 

576 

1008 
2116.2 
8640 
11520 
14400 
23040 
28800 

+    Vf 
0 

-  Jt» 

1 

2 

3 

4 

5 

6 

7 

The  agreement  is  good,  and  if  the  pressures  were  ex- 
pressed in  pounds  per  square  inch,  the  whole  numbers 
would  agree  to  within  one  unit. 

The  measurements  of  superheated  steam  are  few  in  num- 
ber ;  but  using  those  given  by  Him,  we  have  the  follow- 
ing: 


418 


AI)[»KM>\. 

>TK.\M 


•ii  MI*  fr«-     fii.  f». 
I*T  kilo.     |*T  Hi. 


1  -', 
»  III 


M  i«:, 


<          7  |U 


141 
Ml 


474.H 
101,0 


74«4« 


PrMTOfM,  Ib*.  per 
Kf|iur*  foot. 


Errorn 


(  ompntad  T.,,,,.!.,. 
l.-l.ul.ir. 


214«         2110.2     -f-^ 


K  »::,{» 


•     + 


Thcuc  reunite  rlo  not  agree  »o  nearly  as  for  satiint. ••! 
and  the  errors  are  all  in  one  .-<-ii-<-.  tin- 
ln-iii^  ;ill  t/»o  lar^e.  This  we  would  ;intiri|»;it«- 
the  more  the  steam  in  Htijierheatcd,  the  more  iii-;ulv  will 
it  behave  like  a  perfect  gan,  awl  conform  nmn-  nt-;irly  to 
c<jtmtion  (H52),  page  172;  or  to  one  of  that  form. 

In  a  pajHir  read  before  the  Society  of  Mechanical 
n-'t-r-  in  May,  ll^Ml),  I  (limMiHHed  neveral  forniH  of 

in  rated  Htcam.     Among  them  wan  the  following,  J 
|i- .-»••!  by  Mr.  K.  It.  DuwHon,  a  gnidnate  at  the  Institute  : 


=  0.0784  -  - 


-  0.16. 


The  clone  agreement  of  Zeuner's  equation,  (22)  abo\-«-, 
with  ex ftcritncnt,  is  more  apparent  than  real;  since  if  our 
values  of  ./ =  427  instead  of  424,  and  r0  =  -  273.7  in- 
stead of  -  278  be  used,  the  computed  results  won  1*1  IK; 
more  in  error  than  his  appeared  to  be.  I  would  UM-  tin; 
preceding  liquation,  (84),  with  as  much  confidence  for  super- 
lic.iird  steam  as  I  would  Zeuner'g. 

78a.  The  critical  temperature  is  the  state  at  which  710 
external  work  is  done  when  a  liquid  changes  to  a  vapor.  ( )r, 
in  a  series  of  isothermal  changes  in  which  the  li<|iii<l  is 
changed  to  a  vapor,  it  is  the  ,-tatr  at  which  the  external 
work  vanishes.  Or,  again,  it  is  the  state,  at  which  the  /•-  «l 
latent  lieat  of  evaporation  i>  /rro. 


ADDENDA.  419 

Page  104,  the  external  work  done  during  the  evaporation 
of  one  pound  of  the  liquid  will  be  p  X  B  C,  at  the  tem- 
perature T.  When  the  abscissa  between  F  E  and  E  C  is 
zero,  as  at  E,  the  corresponding  temperature  will  be  the 
critical  temperature  for  that  substance. 

Neglecting  the  external  work  due  to  the  enlargement  of 
tin1  volume  of  the  liquid,  the  critical  temperature  will  be 
that  temperature  which  will  reduce  the  apparent  as  well  as 
the  real  latent  heat  of  evaporation  to  zero.  For  steam,  the 
first  two  terms  of  equation  (12)  of  this  Addenda  gives 

r  =  1919  ;  or  T  —  1459°  F. 

The  critical  temperature  of  a  few  substances  has  been 
found  by  experiment.  Thus,  it  is  for 

Deg.C. 

Carbon  tetraehloride 292.5 

Carl>on  disulphide 270.1 

Acetone 240.1 

(Fogy.  Ann.,  cli.,  (1874),  303.) 

Theory  gives  higher  values  than  these.  The  critical 
temperatures  and  pressures  for  twenty-one  substances  is 
given  iu  the  Philosophical  Magazine,  1884,  (2\  page  214. 

Avenarius  showed  by  experiment  that  over  a  certain 
temperature  fixed  for  each  substance  there  is  no  distinction 
between  the  liquid  and  vapor  states,  so  that  pressure  alone 
will  not  cause  a  gas  to  liquefy. 

79.  In  Rankine's  tables  the  absolute  zero  was  assumed 
at  401.2  below  the  zero  of  Fahrenheit's  scale,  while  those  at 
the  end  of  this  work  are  computed  with  400.00. 

85.  See  Article  75  of  this  Addenda. 

9(>.  Clausius  claims  to  be  the  first  to  announce  that  in- 
ternal work  is  a  function  of  the  initial  and  terminal  states 
onJy.  (Clausius  On  Heat,  page  •"••">.) 


420  ADDENDA. 

97.  It  will  be  a  good  exercise  for  the  student  to  give 
geometrical  interpretations  of  the  equations  on  page  132. 

97a.  "  On,  the  dimensions  of  temperature  in  length,  mass  and  time ; 
and  on  the  absolute  C.  G.  S.  unit  of  temperature  "  (Phil.  Mag.,  1887,  (2), 
96).  It  is  shown  that,  in  accordance  with  Thomson's  absolute  scale,  the 
unit  temperature  would  be  that  of  a  perfect  gas  whose  mean  kinetic  energy 
per  molecule  was  one  erg. 

If  E  =  the  mean  kinetic  energy  of  a  molecule  of  the  gas, 
T  =  absolute  temperature  of  a  perfect  gas, 
k  =  a  constant, 
p  =  pressure  per  unit, 
«  =  volume  of  a  pound  of  the  gas, 
n  =  the  number  of  molecules  in  a  pound  ; 
then, 

E  =  k  r, 

and  making  k  =  1, 
E=  r. 
But,  equation  (2)  of  Appendix,  gives,  if  t>,  =  |  for  1  pound, 

xp  f  i  =  t>*  =  2  «.  E; 

hence,  making  x  =  3,  and  omitting  the  subscript,  we  have 
p  v  =  §  n  E ; 

•••*=*  OT  =  2-5xl°-''  =  r 

for  the  value  of  the  temperature  at  0°  C.,  or  —  273"  C.  absolute,  C.  G.  8., 
and 

tfie  absolute  unit  =  273  -*-  2.5  X  10-'*  =  about  1018  C.  degrees. 

"  Having  seen  that  temperature  is  of  the  same  dimensions  as  energy, 
and  knowing  that  the  same  is  true  of  heat,  it  follows  that  entropy,  whose 
dimensions  are  heat  -f  temperature,  is  a  purely  numerical  quantity  ;  and 
the  unit  of  entropy  is  therefore  independent  of  all  other  physical  units. 
In  fact,  the  entropy  of  a  perfect  gas  increases  by  unity,  when  (without  alter- 
ing its  temperature)  it  receives  by  conduction  a  quantity  of  heat  equal  to  the 
mean  energy  of  one  of  its  molecules." 

98.  "Priming  or  superheating.     Equation  (135), 
page  144,  may  be  put  in  a  more  customary  form  as  fol- 
lows : 

Let 

T,  =  the  temperature  of  the  water  at  the  boiling  point 
under  the  given  pressure  in  degrees  Falir., 


ADDENDA.  421 

T^  =  the  temperature  of  the  feed  water,  which  is  as- 
sumed to  be  the  same  as  that  of  the  higher 
temperature  of  the  water  supplying  the  calo- 
rimeter. 

T3  =  the  initial  temperature  of  the  water  supplied  to 
the  calorimeter. 

h  =  the  total  heat  of  steam,  being  the  heat  units 
necessary  to  raise  the  temperature  of  one 
pound  of  water  from  32°  F.  to  the  boiling 
point  and  evaporating  it  at  that  point.  Its 
value  may  be  found  in  special  tables,  or  from 
equation  (13-i)  after  making  x  =  1.  * 

h1  =  the  heat  units  in  one  pound  of  the  water  at  the 
boiling  point  above  32°  F.,  which  will  be  T,  - 
32,  nearly,  but  may  be  found  more  accurately 
from  Article  95,  or  more  directly  from  suit- 
able tables  : 

A,  =  the  heat  units  in  one  pound  of  the  feed  water 
above  32°  F., 

h  —  the  heat  units  in  one  pound  of  the  steam  above 
the  temperature  of  the  feed  water  as  deter- 
mined from  a  calorimeter, 

w  =  the  weight  of  the  steam  condensed  in  the  calo- 
rimeter, 

W  =  the  weight  of  the  water   supplied   to   the  calo- 

rimeter ; 
then 

*=        (Tt-T,\  (1) 


7<e  =  h  -  h,    =  h  -  (T>  -  32).      (3) 

Dividing  both  numerator  and  denominator  of  equa- 
tion (135)  by  J  =  C,  and-  substituting  the  above  values 
gives 


422  ADDENDA. 

Per  cent  of  priming  =  100     e  +  TI  ~  TQ  ~  (4) 


=  100 


h  --  /<, 
If  this  equation  becomes  negative  there  will  have  been 

superheating. 

In  equation  (5),  li  —  A,  is  the  heat  supplied  above  the 

temperature  of  the  feed  water  to  produce  one  poun<I  of 

saturated  steam,  and  —  (Tt  —  Tt)  is  the  heat  supplied  to 

one  pound  of  steam  and  water  combined.  If  the  entire 
pound  be  steam  at  the  point  of  saturation,  this  quantity 
would  equal  the  former  and  the  expression  would  reduce  to 
zero.  But  if  it  exceeded  the  former  there  would  have  been 
superheating,  and  the  expression  becomes  negative.  The 
numerical  difference  between  the  two  terms  would  be  the 
number  of  heat  units  of  superheating,  and  this  divided  by 
the  specific  heat  of  steam  (0.48)  gives  the  number  of  de- 
grees of  superheating  ;  or 

II        2  (6) 

Deg.  superheating  =  —          — j^^ 

These  two  expressions  may  be   partly    combined  in  one, 
thus  ;  if 


Per  cent  priming  =  100  j- 


then,  if  q  be  positive,  we  have 

Per  ce 
and  if  q  be  negative, 

Degrees  of  superheating  =  -  =  ^-|g 
disregarding  the  sign  of  q  in  the  result. 


ADDENDA.  423 

Accuracy  in  the  calorimeter  test  requires  a  correction  for 
the  heat  transferred  to  the  vessel  and  the  vapor  escaping  at 
the  surface.  The  corrections  will  depend  upon  the  size  of 
the  vessel,  its  material  and  the  manner  of  conducting  the 
experiments,  and  do  not  admit  of  a  general  practical  ex- 
pression. 

99.  Specific  heat  of  saturated  vapor.  To  ex- 
plain more  fully  the  fact  that  the  specific  heat  of  saturated 
steam  is  negative,  let  c  A  I,  Fig.  <?,  be  the  curve  of  satu- 
ration, and  let  A,  on  this  curve,  be 
the  state  of  the  steam ;  then  it  is 
found  that  if  it  expands  without 
transmission  of  heat  the  curve  of 
pressures  will  fall  below  A  b  and 
follow  some  line  as  A  d  (equations 
(150)  and  (151),  page  154,  and  below 
equation  (185«),  page  178).  Some  of 
the  steam  will  have  been  condensed, 
so  that  there  will  be  a  less  mass  of  vapor  than  at  the  state 
A.  In  order,  then,  that  the  initial  amount,  one  pound, 
shall  remain  as  steam  during  the  expansion,  heat  must  be 
supplied  to  the  steam.  'On  the  other  hand,  if  the.  steam 
be  compressed  without  transmission  of  heat  it  will  be  su- 
perheated and  follow  some  line  as  A  e ;  and  to  make  its 
path  coincide  with  A  c,  heat  must  be  abstracted.  Rankine 
found,  from  an  inspection  of  actual  curves  of  expansion 
as  shown  on  indicator  diagrams,  that  the  equation  of  the 
curve  of  expansion  was 

p  v**  —  constant,  nearly. 

But  we  have  previously  found  that  the  equation  of  the 
curve  of  saturation  is 

p  -yil  =  constant,  nearly ; 
which  equations  show  that,  if  they  have  one  common  state. 


424  ADDENDA. 

the  curve  of  expansion  will  fall  under  the  curve  of  satura- 
tion, and  the  curve  of  compression  rise  above  it,  since  it  is 
a  steeper  curve  at  the  common  point.  This  is  true  for  all 
temperatures  of  steam  less  than  about  976°  F. 

Ether  engine.     The    specific  heat  of  the  saturated 
vapor  of  ether  will  be  equation  (139),  page  147, 


and  retaining   only  two  terms   of  equation   (12),,    of   this 
Addenda,  we  have 

Ae  =  93.32  +  0.3870  r  ; 


0.3870; 


.. 

-j-5  =  0.3870. 
d  r 

also  have 

c  =  0.517  ; 

. 

which  is  zero  for  '» 

f  =  180.5°  ; 

or,  T  =  -  280°  F  ; 

hence,  for  all  temperatures  above  —280°  F.,  s  will  be  posi- 
tive ;  that  is,  the  specific  heat  of  the  saturated  vapor  of  ether 
is  positive  for  all  temperatures  occurring  in  practice. 

This  reverses  certain  conditions  of  the  steam-engine. 
Eeferring  to  Fig.  c,  now  let  e  A  d  be  the  curve  of  satu- 
ration of  the  vapor  of  ether  ;  then  if  the  vapor  be  com- 
pressed from  the  state  A  the  path  of  the  fluid  will  fall 
below  A  e,  and  heat  must  be  supplied  in  order  that  the 
path  shall  be  A  e,  that  of  saturation.  On  the  contrary,  if  it 
be  expanded  without  transmission  of  heat,  the  vapor  will  be 
superheated  and  its  path  will  rise  above  A  d  ;  and  in  order 
that  its  path  shall  coincide  with  A  <7,  heat  must  be  ab- 
stracted. 


ADDENDA.  425 

In  the  ether  engine,  if  v^  be  the  volume  of  a  pound  of  the 
saturated  vapor  at  a  temperature  r2,  and  it  be  compressed 
until  its  temperature  is  T,  then  will  the  volume  of  the  vapor 
be,  equation  (150),  page  154, 


and  the  volume  of  the  vapor  condensed  will  be 


Other  vapors  may  be  examined  in  the  same  manner.  (See 
Vapor  Engines  in  the  following  Article.) 

Temperature  of  inversion.  The  temperature  at 
which  the  specific  heat  of  the  vapor  becomes  zero  is  called 
the  temperature  of  inversion  /  below  which  expansion  will 
cause  condensation,  and  above  which  expansion  will  cause  va- 
porization, and  the  contrary,  the  specific  heat  being  negative. 

"  M.  Him*  gave  an  experimental  demonstration  in  1862  of  the  con- 
densation accompanying  a  sudden  adiabatic  expansion  of  dry  saturated 
steam,  and  thus  proved  the  negative  sign  of  *'  at  low  temperatures  ;  he 
allowed  steam  to  pass  gently  from  a  boiler,  where  it  was  generated  under 
5  atmospheres  pressure,  through  a  copper  cylinder,  200  c.  long  and  15  c. 
in  diameter,  the  ends  of  which  were  closed  by  plates  of  glass,  until  all 
air  and  condensed  water  had  been  driven  out  and  the  sides  had  attained 
the  temperature  of  the  steam  ;  the  exit-stopcock  of  the  cylinder  was  then 
shut,  and,  the  cylinder  being  full  of  dry  saturated  steam,  the  connection 
with  the  boiler  was  cut  off  and  the  exit-stopcock  suddenly  opened  ;  the 
pressure  at  once  fell,  and  the  cylinder,  which  had  previously  appeared 
perfectly  transparent  to  an  observer  looking  along  its  axis,  became  per- 
fectly opaque  from  the  formation  of  a  cloud  ;  this  cloud,  however,  soon, 
disappeared,  heat  being  supplied  by  the  vessel  as  it  cooled  from  152°  C. 
(the  temperature  of  saturated  steam  under  5  ordinary  atmospheres)  to 
100°  C.  (the  temperature  under  1  atmosphere). 

M.  Cazinf  improved  this  apparatus  by  connecting  the  cylinder  with 
another  in  which  a  piston  was  movable,  and  placing  the  whole  in  an  oil- 
bath  the  temperature  of  which  could  be  varied  at  will ;  saturated  vapor 
in  the  one  could  then  be  allowed  to  suddenly  expand  into  the  other,  or 

*  Bulletin  de  la  Soc.  industr.  de  Mulhouse,  cxxxiii.  (1862),  129  ;  Cosmos,  xxii.  (J863), 
413. 
t  C.  B.  Ixii.  (1866),  56. 


426  ADDENDA. 

•when  filling  both  could  be  suddenly  compressed  by  the  motion  of  the 
piston.  A  cloud  was  always  formed  when  steam  expanded,  but  never  on 
its  compression,  and  with  carbon  bisulphide  the  same  occurred  ;  on  the 
contrary,  as  noticed  earlier  by  M.  Him,  ether  vapor  never  condensed 
during  expansion,  but  always  on  compression,  showing  that  its  temper- 
ature of  inversion  is  below  ordinary  temperatures  ;  further,  the  temper- 
ature of  inversion  appeared  to  be  between  125°  and  129°  C.  for  chlo- 
roform, and  for  benzine  between  115°  and  130°  C.  These  results  arc  in 
keeping  with  theory,  M.  Regnault's  formulae  giving  790.2°,  —  113°, 
123.5C,  100°  C.  for  the  temperatures  of  inversion  of  carbon  bisulphide, 
ether,  chloroform,  and  benzine  respectively." 


The  results  of  Articles  110,  111,  112,  illustrate  the 
fact  that  the  efficiency,  when  the  operation  is  in  the  cycles 
there  assumed,  is  less  than  that  for  Carnot's  cycle  when 
worked  between  the  same  limits  of  temperature.  The  in- 
itial pressures  are  the  same  in  each  of  the  three  cases,  being 
Pi  =  1-4400  Ibs.  The  range  of  temperatures  in  the  Exer- 
cise on  page  174,  for  superheated  steam,  expanded  iso.ther- 
mallv,  is 

T,  -  T<  =  450  -  110  =  340. 

Initial  absolute  temperature  =  910.66  ; 

hence,  if  the  heat  absorbed  had  been  worked  in  a  Carnot's 
cycle  the  efficiency  would  have  been 


instead  of  0.207  as  given  in  equation  (171a). 

In  the  Exercises  on  pages  176  and  178  for  saturated  steam 
expanded  adiabatically,  the  initial  and  final  temperatures  are 
the  same,  being 

Tt  =  327,6,  T<  =  110  ; 

.  •  .  T,  -  Tt  =  217.6, 
T,  =  460.66  +  327.6  =  778.66  ; 

T  —  T 
.  •  .  ti  -  ±1  =  0.267, 

instead  of  0.204  or  simply  0.200.     The  latter  is  75  per  cent 
of  the  former. 


ADDENDA. 


427 


If  the  feed  water  were  of  the  same  temperature  as  the 
exhaust  steam,  we  would  have 

T,  -  T3       327.6  -  134 


G 
PIG.  d. 


H 


which  also  exceeds  the  efficiencies  actually  found. 

Vapor  engines.     To  find  the  maximum  efficiency  of 
the  ideal  vapor  engine. 

Let  M  0  =  VA,  Fig.  d,  be  the  volume  of  one  pound  of  the 
liquid,  which  assume  to  be  constant  at  all  pressures,  A  0  =  p^ 
=  the  absolute  pressure  at  the  temperature  Tr     Let  the  vapor 
expand,     the    temperature     being 
maintained  constant ;  then,  since  it      D    A      B 
is  in  contact  with  the   liquid,  the 
vapor   will   be   saturated    and   the 
pressure  will  be  constant,  equation 
(80),   and  the  line  A  B,  parallel  to 
O  II,  will  be  an  isothermal.     Con- 
tinue the  expansion  until  the  liquid 
is  completely  evaporated,  and  rep- 
resent the  volume  Jf  G  by  VB.  From 

B  let  it  be  expanded  without  transmission  of  heat  to  a  tem- 
perature T^  and  volume  M II  =  vc ;  then  compressed,  ab- 
stracting heat  so  as  to  maintain  the  constant  temperature  rn 
and  pressure  pv  to  such  a  point  J  that  when  compressed  with- 
out transmission  of  heat  it  will  be  entirely  liquefied  when  it 
reaches  the  initial  state  A.  The  adiabatics  B  C  and  A  J 
will  be  represented  by  equation  (149).  The  line  B  N  rep- 
resents the  curve  of  saturation,  and  if  the  specific  heat  of 
the  vapor  be  negative,  as  steam,  the  adiabatic  B  C  will  fall 
under  the  line  B  N  as  before  shown.  But  if  the  specific 
heat  be  positive,  like  the  vapor  of  ether,  then  may  B  C  rep- 
resent the  curve  of  saturation  and  B  N  an  adiabatic,  pro- 
vided the  vapor  is  saturated  at  state  JV ;  or,  in  other  words, 
if  the  vapor  is  to  remain  saturated,  the  expansion  A  B  will 


428  ADDENDA. 

extend  to  such  a  point  B  that,  when  expanded  adiabatically, 
reducing  the  temperature  from  rl  to  ra,  the  liquid  will  not 
be  completely  evaporated  before  the  lower  temperature  is 
reached.  If  the  ether  be  completely  evaporated  at  state  B 
it  will  be  superheated  when  expanded  adiabatically  as  shown 
in  Article  99  of  this  Addenda,  and  B  -2V  will  be  the  adia- 
balic  of  an  imperfect  gas.  Similarly,  the  point  of  com- 
pression J  will  be  further  to  the  right  for  ether,  and  all 
vapors  whose  specific  heat  is  positive,  than  for  those  which 
are  negative. 

The  heat  absorbed  will  be,  equations  (21)  and  (74), 


*•  ='' 

The  heat  emitted  from  C  to  «/will  be 

//,  =  T,  *&  K  -  v,).  (2) 

Making  d  r^  =  d  T,  =  d  r,  we  have, 

ABcd=    ^V*.  -  <O  ==  ^  («o  -  *,),         (3) 

since  the  areas  of  all  the  strips  formed  by  equidistant  iso- 
thermals  between    adiabatics  will    be  equal,  as  shown  in 
Article  40,  page  32. 
The  efficiency  will  be 

.  -  ff.       (^-O^K-^)  =  rLj-jri> 


which  is  the  same  as  for  the  perfect  elementary  engine, 
equation  (159),  page  161.  This  result  might  have  been 
anticipated,  since  the  cycle  is  Carnot's. 

97«.  To  represent  geometrically  certain  rela- 
tions. 

Equations  (A),  page  48,  or  (123),  page  135,  give 


ADDENDA 

d  II      (dp 


429 


dr 

--  . 

r 


If  r  be  constant  during  the  expansion  d  v,  we  have  d  r  — 
0,  and 


Dividing  both  sides  by  d  v  gives 


d_v\    _  fdp\ 
dvJr~  \d  r); 


The  factor  (-=2-  I   is  the  rate  of   change  of  pressure  per 

\d  T/V 

unit  of  temperature,  and  therefore  if  the  rate  were  uniform 
during  the  change  of  unity  of  temperature  from  r  to  t  -f-  1, 
it  would  be  the  increase 
of  pressure  due  to  an  in- 
crease of  one  degree  of 
temperature.  Draw  two 
parallel  lines  to  represent 
two  isothermals  differing 
by  unity.  As  in  the  cal- 
culus, these  lines  may  be 
tangents  to  actual  isother- 
mals. Let  a  w  and  g  p  be  a 
the  isothermals,  differing 
by  one  degree  of  tempera- 
ture. At  a  let  the  pres- 
sure be  p  ;  then  will 

dp\ 

-^-  ]  =  a  c. 

d  T  /v 

If  the  abscissa  of  J  in  ref- 
erence to  a  be  a  e  =  d  v, 

then  a  cfb  —  ac  •  ae  =  l-JL\   d  v. 


430  ADDENDA. 

Let  the  straight  line  a  g  through  a  be  an  adiabatic  (tangent 
to  an  actual  adiabatic).  Divide  a  p  into  parts  each  equal  to 
d  i),  and  through  the  points  of  division  draw  lines  parallel 
to  a  c,  and  through  their  points  of  intersection  with  a  w 
draw  lines  parallel  to  a  g  ;  the  spaces  thus  formed  will  be 
equal  ;  and  equal  to  a  cfl,  since  they  have  the  same  base 
and  altitude.  Hence 


Dividing  by  d  v, 


Let  a  i  represent  unity  of  volume,  or 

a  i  =  1  ; 
then 

id 


which  is  the  right  member  of  the  preceding  equation. 

f  -~J  is  the  number  of  d  <p's  between  r  and  r  -f-  1  for  an 

increase  of  volume  equal  to  unity,  and  is  equivalent  to  the 
area  a  g  z  m  =  a  c  y  vn>\ 

ldp\ 


as  deduced  above,  analytically. 

It  will  be  observed  that  the  subscript  of  the  parenthesis 
on  one  side  of  the  equation  is  the  same  as  the  independent 
variable  on  the  other  side ;  that  is,  r  is  a  subscript  on  the 
left  side,  and  the  independent  variable  on  the  other.  Writ- 
ing the  reciprocal  of  the  preceding  equation,  and  observing 
a  similar  order  of  subscripts,  we  have, 


(d  T\    f 

-  135V 


ADDENDA.  431 

the  geometrical  interpretation  of  which  we  will  proceed  to 
show. 

The  equation 

<p  =  a  constant 

is  the  equation  of  an  adiabatio,  and  the  expression  I — 

implies  the  rate  of  change  of  temperature  due  to  a  change 
in  the  pressure  along  the  adiabatic  a  t.  Let  the  pressure  at 
a  be^>,  and  at  t,  p  +  1,  and  a  q,  parallel  to  the  axis  of  vol- 
umes ;  then 

qt  =  l. 

The  temperature  at  a  being  T,  that  at  t  will  be  r  -f-  I  y-  U; 

\ftpi 

.  • .  change  of  temperature  from  a  to  t  =  ( -=—  }  . 

\dph 

The  area  aghb  =  dcp;  and  if  this  area  were  extended 
to  t,  its  value  would  be 


and  this  also  equals 


tpb 

Assume  any  arbitrary  area  as  a  g  I  k  for  an  increase  of 
(p  =  1.      Since  I  -j — )  is  the  increase  of  temperature  for  an 

increase  of  unity  of  pressure,  it  is  the  temperature  of  the 
isothermal  t  o  above  that  of  a  k ; 

.  • .  area  a  t  n  k  =  ( -, — j    ((p  =  1). 

The  expression  (-3 — j  is  the  abscissa  of  v  for  (p  unity,  and 

hence  is  the  abscissa  of  k  in  reference  to  a ;  and  this  multi- 
plied by  q  t  =  1,  gives 


432  ADDENDA. 


(dv 
1  .     -y—    =  atn 


hence  the  equality  of  the  expressions. 

The  second  equation  of  thermodynamics  is 

,  d  II       n    dr 

*<?  =  —   =C,— 

which  becomes  for  r  constant, 


If  a  and  b  be  consecutive  states  on  the  isothermal,  for  an 
increase  of  volume  d  v  =  a  e,  the  pressure  decreases  an 
amount  a  d  =  —  dp. 

(dv 


where  ap  is  the  increase  of  volume  in  passing  from  one 
isothermal,  r,  to  another  a  unit  higher,  r  -\-  1,  measured 
on  a  horizontal  line  ;  and  this  multiplied  by  an  ordinate  q  t 


=  p  =  1,  gives  the  area  a  t  up  =  a  t  v  w. 

number  of  <p's  between  two  isothermals  for  an  increase  unity 
in  j?. 

d  tpT  —  ab  h  g  =  a  bjjp, 

which  divided  by  —  dp  gives 

—  a  tup  =  -  &£}    =  atvw. 
\dp  Jr 

Taking  the  reciprocal,  we  have 

fdp_\    =  _    (dr\ 
\  d  q>/'v  \dv  /*' 

which  may  be  geometrized  in  a  similar  manner. 


ADDENDA.  433 

Thus,  the  right-hand  member  implies  the  increase  of 
temperature  for  a  change  of  unity  of  volume  on  an  adia- 
batic.  Let  a  q  =  —  v  =  —  1,  and  from  q  erect  a  perpen- 
dicular to  meet  the  adiabatic  through  a  at  t  ;  then  will  the 
increase  of  temperature  at  t  above  that  at  a  be 


<4  v,  . 
If  the  increase  unity  of  cp  be  represented  by  a  g  Ik,  then 


Also 

/d\ 

\d 


/dj\ 

\d  <pj 


will  be  the  ordinate  of  g  above  a,  which  multiplied  by  v  = 
unity  will  give  the  same  result  as  the  preceding. 

113.  Cut-off.  In  the  analysis  on  page  182  no  account 
is  taken  of  the  condensation  of  steam.  In  order  to  determine 
approximately  the  modification  of  the  preceding  analysis 
which  would  result  from  this  cause,  assume  that  sufficient 
steam  is  admitted  to  produce  the  same  pressure  up  to  the 
point  of  cut-off  as  if  there  were  no  condensation,  and  that 
the  mean  effective  pressure  is  the  same  ;  then  will  all  the 
quantities  in  expression  (191),  page  183,  remain  the  same 
except  r  and  h.  The  value  of  h  will  be  increased,  for  larger 
boilers  may  be  required,  and  more  fuel  will  be  required, 
and  other  incidentals  may  be  increased.  In  the  solution,  if 
h  be  increased,  O  C,  Fig.  50,  page  184,  will  be  increased, 
and  C  D  will  be  increased  in  like  proportion,  since 

CD  =  -  x  o  a 

Hence,  the  point  D  will  be  found  on  the  same  line 
-2V  O  D,  prolonged  ;  but  the  more  remote  the  point  D  is, 
the  further  to  the  right  will  be  the  tangent  point  P,  thus 
requiring  a  later  cut-off.  In  theory  or  practice,  however, 


434  ADDENDA. 

the  cut-off  in  case  of  condensation  will  not  be  very  much 
larger  than  if  there  were  no  condensation.  This  cast-  was 
well  discussed  in  the  paper  by  Wolff  and  Denton  in  their 
paper  read  before  the  Society  of  Mechanical  Engineers,  re- 
ferred to  on  page  189.  It  appears  from  the  examples 
there  given,  that  the  first  cost  has  an  important  influence  on 
the  most  economical  point  of  cut-off,  and  that  in  stationary 
engines  the  ratio  of  expansion  should  not  be  very  high  for 
greatest  economy — all  elements  involved  being  considered. 
But  in  marine  engines  another  element  enters — that  of 
storage  room  for  fuel.  Here,  all  space  saved  by  reducing 
the  amount  of  coal  necessary  for  making  a  voyage  may  be- 
come profitable  by  furnishing  room  for  the  storage  of  mer- 
chandise, or,  otherwise,  lessening  the  non-paying  burden  of 
the  ship.  Under  these  conditions  very  high  expansions  may 
be  economical  when  they  would  not  be  on  land  service, 
where  storage  room  is  of  no  value. 
Miscellaneous. 

The  following  are  some  average  results  for  the  total  heat  of  combustion 
per  pound  of  some  liquid  fuels.  (Thesis,  by  James  Beatty,  Jr.,  Stevens 
Institute.  1884,  p.  72.) 

Petroleum 20,300  B.  thermal  units. 

Coal  gas 20,200" 

Creosote 16,400"  '     " 

Asphalt 15,900"  " 

Anthracite 14,500"  '• 

Carbon 14,450"  «' 

Bituminous   14,200"  «* 

Lignite 11,800"  .      " 

Peat 9,600"  " 

Water  gas 8,500  "  " 

Wood 6.800" 

Generator  gas 3,100"  " 

Air  liquefied  at  high  pressures — say  with  about  40  atmospheres  and  at 
—  142°  C. — may  be  maintained  in  a  liquid  state  at  a  pressure  of  one 
atmosphere,  provided  the  temperature  be  lowered  to  about  —  191°  C. 
(Phil.  Mag..  1885,  (2),  463.) 

This  u  similar  to  the  action  of  steam.     If  steam  be  at  a  pressure  of 


ADDENDA.  435 

200  pounds  to  the  square  inch  and  temperature  390°  F.,  it  cannot  be 
liquefied  by  mere  compression  ;  but  if  at  that  pressure  the  temperature 
be  diminished  to  380°,  it  will-liquefy,  and  the  pressure  may  be  gradually 
removed  down  to  one  atmosphere,  provided  the  temperature  be  reduced 
to  212°  F.  or  less. 

Curve  of  boiling  points  of  oxygen.  The  experiments  of  Wroblewski  for 
the  boiling  points  of  oxygen  give  the  following  results  (Phil.  Mag., 
1884,  (1),  491) : 

Pressure.  Temperature. 

Atm.  Degrees  C.  Degrees  F. 

50  —  113  - 171.4 

27.02  -  129.6  -  201.3 

25.85  -131.6  -204.9 

24.40  -  133.4  -  208.1 

23.18  -  134.8  -  210.6 

22.2  -  135.8  -  212.4 

1  -184.0  -299.2 

These  plotted  on  the  plane  of  p  and  T  would  be  the  curve  of  saturation 
on  that  plane. 

Evaporation  and  dissociation.  (Phil.  Mag.,  1887,  (2),  196. 
This  is  Part  IV.  of  a  series  of  articles  upon  liquids  and  vapors.) 

Boiling  points  of  gases  at  atmospheric  pressure,  or  the  points 
of  liquefaction  of  gases.  M,  S.  Wroblewski  has  determined  the  follow- 
ing values,  the  temperature  being  measured  by  means  of  a  thermo- 
electrical  apparatus  which  was  considered  correct  to  within  ^  of  a  degree 

down  to  —  200°  C. 

Boiling  Point  at 
Substance.  Atmospheric  Pressure. 

Degrees. 
C.  F. 

Air -194.2        -317.5 

Oxygen,  O -184.0        -299.2 

Nitrogen,  N -  193.1        -  315.6 

Carbonic  oxide,  CO -  193.0        -  315.4 

Ethyline -  193.0        -  315.4 

(Phil.  Mag.,  1884,  (1),  490.) 

The  conductivity  of  heat  in  several  liquids  is  given  in  the  Phil.  Mag., 
1887,  (2),  1-27.  The  conductivity  of  water  is  the  highest  of  all  those 
reported.  If  that  of  water  be  100,  that  of  benzine  is  about  2P.8,  p.  18. 
If  that  of  air  be  0  00049  C.  G.  S.,  that  of  water  will  be  0.0645,  p.  22 ; 
and  that  of  petroleum  about  0.022 ;  ether,  0.023  ;  carbon  disulphide, 
0.016,  p.  27.  Accordingly,  it  appears  that  water  is  one  of  the  best  of  the 
liquids  for  absorbing  heat  in  a  boiler. 


436  ADDENDA. 

Watt's  law.  The  sum  of  the  sensible  and  latent  heats  of  steam  is 
constant. 

This  is  one  of  the  earlier  laws  deduced  from  crude  experiments,  but 
Regnault's  experiments  showed  it  to  be  erroneous.  (Eq.  (78),  p.  95. 
Also  Browne's  translation  of  Clausius  On  Heat,  pp.  131-134.) 

The  mechanical  equivalent  of  light,  as  determined  by  Ju- 
lius Thomsen,  is  about  12.28  foot-pounds  per  minute  for  one  caudle. 
(Phil.  Mag  ,  1865,  Vol.  XXX.,  p.  246.) 

According  to  Moses  G.  Farmer,  it  is  about  10.1  foot-pounds  per  candle 
per  minute.  (Am.  Jour.  Sc.  and  Arts,  1886,  Vol.  XLL,  p.  214.) 

Density  of  liquefied  gases.  The  density  of  liquid  hydrogen  at 
0°  C.  is  0.025  ;  at  -  23°  C.,  0.032  ;  of  liquid  nitrogen  at  0°  C.,  0.37  ;  of 
liquid  oxygen,  0.65  at  0°  C.  (Caillett  and  Hautefeuelle  in  Comptes  Ren- 
dus,  Vol.  XCIL,  p.  1086.) 

According  to  Wroblewski,  the  density  of  liquefied  oxygen  is  0.899  at 
—  100°  C.,  and  1.24  at  -  200°  C.  (Comptes  Rendus,  102,  1010.) 

The  specific  gravity  of  ice  at  0°  C.  is  0.91886.  (Smithsonian 
Contributions,  1888,  Vol.  XXXIII.,  p.  40.) 

Sulphurous  acid.  Latent  heat  of  evaporation  of  sulphurous  acid 
is 

ht  =  91.2  -  0.37  t  calories, 

very  nearly  from  t  =  -  10°  C.  to  t  =  70°  C. 

Density  of  the  saturated  gas  is  0.00624  at  7.3°  C.;  0.4017  at  154.9°  C. 

Density  of  the  liquid,  1.4338  at  0°  C.;  0.6370  at  155.05°  C.  (Comptes 
Rendus,  Vol.  CIV.,  p.  1564.) 

For  more  recent  and  more  reliable  values,  see  page  357,  and  paper  by 
the  author  in  the  Trans,  of  the  Society  of  Mechanical  Engineers,  Vol. 
XI.,  1890. 

Short  trips  across  the  Atlantic.  In  May,  1889,  the  City  of 
Paris  crossed  westward  in  5  days,  23  hours  and  7  minutes.  In  August, 
eastward,  in  5  days,  23  hours  and  44  minutes  ;  westward,  in  5  days,  1'J 
hours  and  18  minutes.  In  August.  1890,  the  Teutonic  made  a  westward 
trip  in  5  days,  19  hours  and  5  minutes,  and  claimed  to  have  sailed  40 
miles  farther  than  the  City  of  Paris  when  the  latter  made  her  quickest 
trip.  The  greatest  reported  speed  of  the  latter  exceeded  that  of  the 
former  by  5  miles  for  24  hours.  In  August,  1891,  the  Teutonic  made  a 
trip  in  5  days,  16  hours  and  31  minutes.  July,  189 »,  City  of  Paris,  5  days, 
15  hours,  58  minutes. 

Vertical  distribution  of  temperature  in  the  atmosphere  is 
very  variable.  (Engineering  [London],  1889,  November,  pp  522.  566.) 

Temperature  of  space  has  no  sensible  existence.  (S.  P.  Langley, 
Phil.  Mag.,  January,  1890,  p.  33.)  The  moan  tenperature  of  sunlit  lunar 


ADDENDA.  437 

soil  is  much  lower  than  has  been  supposed,  and  is  most  probably  not 
greatly  above  zero  centigrade.     (Ibid,  p.  53.  ) 

Heat  transmitted  through  cast-iron  plates  pickled  in  nitric  acid 
is  about  70  per  cent,  of  clean  untreated  plates.  (Carpenter,  in  Trans, 
Soc.  Mech.  Eng'rs,  1890.) 

History  of  the  gas-engine.  (Scientific  American  Supplement, 
1889,  p.  11,416.) 

Steam-engine  practice  in  1884  by  J.  C.  Hoadley,  C.  E.  (Trans. 
Am.  Asso.for  Ad.  of  Science,  1884,  pp.  289-359.) 

While  breathing,  a  man  consumes  about  215  cubic  feet  of  air  per 
hour,  or  about  17  pounds. 

Page  74.—  To  show  that  the  elasticity  of  the  air  equals  its  tension, 
let  PI  be  the  pressure  on  unity  of  area,  producing  a  compression  A  and  I 
the  length  of  the  column  ;  then,  from  Resistance  of  Materials,  we 
have 


For  isothermal  change  we  have 

p  v  =  R  T; 
.  •  .  p  dv  =  —  v  dp  ; 
or,  making  v  =  I,  dv=—  A  and  d  p  =  PI,  then 


which  compared  -with  equation  (a)  gives 
p  =  E. 
If  the  change  be  adiabatic,  then 

p  J  =  a, 

and  by  a  similar  process  it  will  be  found  that 
E=  yp. 

Page  83.  —  To  obtain  equation  (65),  let  ra  be  variable,  and  dropping 
the  subscript  a,  the  variable  factor  in  (64)  will  be 


ry-i  4/ri-r. 

Squaring,  differentiating,  placing^  equal  to  zero,  and  equation  (65)  ig 
readily  found. 


ADDENDA. 


Page  1O3.  EXERCISE  9— Let  0  v,  be  the  volume  of 
a  pound  of  water  ;  when  it  is  all  evaporated,  let  it  be  0  vt 
=  26.5  cubic  feet.  Let  vl  A  represent  the  pressure  of 

one  atmosphere  =  211G.3 
pounds  per  square  foot; 
then  the  external  work 
will  be  0  f,  X  vt  A.  Let 
vl  a  be  the  virtual  pressure, 
which  would,  if  worked  to 
the  volume  vu  represent 


FIG./. 


both  the  external  and  in- 


ternal work  ;  then,  in  this  exercise,  A  a  will  be  12.37  times 
vt  A.  The  heat  absorbed  in  evaporating  the  pound  of  water 
will  be  represented  by  the  area  q>t  A  B  <?„  which  in  this 
exercise  will  be  751672  foot-pounds. 

The  line  A  B  is  an  isothermal  of  the  vapor  whose  equa- 
tion is  of  the  form 

p  =  o  •  v  +  I. 

The  value  of  p  (or  V)  is  given  by  equation  (80),  page  97 ; 
or  by  the  equations  on  pages  413  and  414.  The  equation 
of  a  I  is  of  the  same  form,  but  for  steam  the  last  term  will 
be  about  11  to  12  times  vt  A.  The  equations  of  the  adia- 
batics  A  <p^  and  Bcp^  will  be  given  by  equation  (149),  page 
153,  or  equation  (a),  page  184. 

Page  1O9.  EXERCISE  1.— Let  t  be  the  required  temperature;  then 
will  the  heat  lost  by  the  water  equal  that  gained  by  the  other  sub- 
stances ;  hence 

3  (90  -  0  =  W(t  -  30)  X  0.655  +  20  (t  -  60)  x  fa  .-.t  =  49.56°  F. 

Another  way  of  solving  these  problems,  when  there  is  no  change  of  state, 
as  from  solid  to  liquid,  is  to  find  the  heat  in  all  the  substances  above 
some  assumed  initial  temperature,  and  divide  the  result  by  the  heat 
necessary  to  raise  the  several  substances  one  degree,  as  follows  : 

Since  the  substances  are  all  liquid  in  this  Exercise,  we  will  treat  them 
as  if  they  were  liquid  from  and  above  0°  F.;  then  will  the  heat  units  in 


ADDENDA. 


439 


3  Ibs.   of  water  at  90°  F.  above  0°  F.  be  3  X  90  X  1 
specific  heat  of  water  is  unity.     In  this  way  find 


Sp- 
h. 


Lb8 
"* 


0.655  x  10     = 
T&  x  20     = 


B.  T.  U. 

per  deg. 
3.000 
6.550 
0.606 


Total.        10.156 


Deg. 
x  90 
x  30 
x  60 


270,   since  the 


B.  T.  U. 
Total. 
270 
196.5 


502.86 


EXERCISE  2.—  la.  this  exercise  there  will  be  a  change  of  state  from  ice 
to  water.     The  heat  units  above  0°  F.  will  be 

heat  in   3  Ibs.  ice  from  0°  to  10°...  .0.5  x     3  x     10  =      15  heat  units 

"     "12    "    solid  water  at  32°....  0.5  x  12  x    32=    192    " 

heat  of  fusion  12  Ibs  ..............  12x144  =  1728    "       " 

heat  in  12  Ibs.  HaO  from  32  to  60°.  .  12  x    28  =    336     "       " 


Total  heat  in  the  mixture  above  0°  F.  =  2271     " 
The  heat  in  the  whole  mass  (15  Ibs.)  above  zero  to  water  at  32°  will 
be 

0.5  x  15  x  32  +  15  x  144  =  2400  heat  units 

This  shows  that  it  would  require  2400  —  2271  =  139  more  heat  units 
to  melt  all  the  ice  at  32°.  If  all  be  melted,  then  the  temperature  of 
melting  would  be  129  -*•  15  =  8.6°  F.  below  32°,  or  23.4°  F.  The  word- 
ing of  the  problem  should  be  changed  in  order  to  obtain  a  consistent 
result.  Thus,  let  it  be  required  to  find  how  much  ice  would  be  melted  ; 
then,  if  y  be  the  pounds  melted,  we  have 

3  (32  —  10)  X  0.5  +  144  y  =  12  (60  —  32)  ; 
.  •.  y  =  2.1. 

EXERCISE  3.—  In  this  there  will  be  two  changes  of  state  from  0°  to 
225°  F.     We  have  for  heat  in  the  two  substances 


6  Ibs.  ice  above  0°  F  ..............  0.504 

1  lb.  HtO  from  0°  to  32°  ...........  0.504 

plus  heat  of  fusion  ................ 

"    from  32°  to  212°  .............. 

"    heat  of  vaporization  .......... 

"    steam  from  212  to  225  ..........  0.48  x 


B.  T.  U. 

6  x    20=      60.48 
1  x     32=      16.128 
1  x  144  =    144. 
1x180=    180. 
1  x  966  =    966. 


13 


5.24 


1371.368 


440  ADDENDA. 

Assuming  that  the  resultant  mixture  is  liquid,  we  next  find  how  much 
heat  would  be  necessary  to  raise  the  temperature  of  7  Ibs.  from  0°  to 
water  at  32°,  we  have 

0.504  x  7  x    32  =    112.896  heat  units. 
7  x  144  =  1008. 


Heat  required  1120.896    "       " 

There  will  remain  1371.368  —  1120.896  =  250.472  thermal  units,  and 
this  would  raise  the  temperature  of  the  7  Ibs.  of  water  250.472  -»-  7  = 
85.75  degrees  ;  hence  the  temperature  will  be  35.75  +  32  =  67.75  degrees 
F.,  and  as  this  is  less  than  212  degrees  it  is  the  result  sought. 

EXERCISE  5.— Let  x  be  the  required  number  of  pounds,  then 
(212  -  200)  x  =  966. 

Page  12O.— In  order  to  reduce  equations  (107)  and  (108),  it  will  be 
necessary  to  find  v  in  terms  of  p  and  r  ;  but  the  equations  for  imperfect 
fluids  usually  being  p  =/  (r,  v),  the  resolution  in  terms  v  will  be  more 
or  less  complex.  Thus,  one  of  the  simplest  forms  is  that  frequently 

used  in  this  work,  p  v  =  a  r  —   — ,  which  resolved  in  terms  of  v  is  a 

radical  of  the  second  degree,  and  the  -z-j  is  so  complex  that  when  multi- 
plied by  d  p  the  integral  cannot  be  performed  in  finite  terms.  If,  how- 
ever, the  equation  be  that  given  by  Zeuner,  equation  (21),  page  414,  v 
will  be  found  as  a  simple  function  of  r  and  p,  and  equations  (107)  and 
(108)  will  be  more  easily  reduced  than  (105)  and  (106). 

Page  123, — To  show  the  geometrical  signification  of 
the  quantities  in  the  equation 


•< 

let  a  I  be  the  path   of  the  fluid. 

Intersect    it  by  two  consecutive 

isothermals,  T  and  r  -f-  d  T,  and 
v  draw  a  d  parallel  to  the  v  axis, 

I  c  and  a  g  parallel  to  the^>  axis. 

Then  the  total  differential  of  v,  or  d  v  (which  is  the  left 
member  of  the  equation),  will  be  the  abscissa  of  b  in  refer- 


ADDENDA.  441 

ence  to  a,  and  the  total  d  p  will  be  the  ordinate  of  I  in 
reference  to  a  ;  hence 

dv  —  a  c,  dp  =  b  c. 

At  a  draw  the  tangent  a  A,  and  let  a  g  =  p  =  1,  and 
make  a  e  =  b  G  and  draw  e  f  and  y  h  parallel  to  a  d,  then 

d 


that  is,  if  the  isothermal  were  the  straight  line  a  h,  then  by 
passing  down  it  until  p  =  1  =  a  g,  the  abscissa  g  h  in  refer- 
ence to  a  will  be  as  written  above.  It  not  being  generally 
a  straight  line,  take  a  e  =  I  c  =  d  p,  then,  by  the  simi- 
larity of  triangles, 


ef=ae=  dp. 

ag  \dp)T    • 

Since  a  e  is  negative,  being  measured    downward,   we 
really  have 


/  \ap  T  j 

by  construction.  This  is  the  first  term  of  the  second  mem- 
ber. In  the  third  term  f  — ?)  would  be  represented  by  a  d, 

VZ  T/P 

since^  is  constant,  provided  the  isothermals  were  unity 
apart ;  but  as  they  are  only  d  r  apart  we  have,  on  the  prin- 
ciple just  given, 


hence, 


442 


ADDENDA. 


If  v  be  constant,  tlie  path  of  the 
will  be  perpendicular  to  the  axis  of  v.  Let 
a  of  the  preceding  figure  be  moved  to  tlie 
right  until  it  falls  on  c,  as  shown  in  the 
annexed  figure,  when  total  d  v  becomes 
zero,  and  we  have 


d 


Page  132.— The  equation 


pjr 


dp  =  —  0  d. 


•was  deduced  from  equation  (A),  p.  48,  by  dividing  both  members  by  d  v  ; 
hence,  in  the  right  member  the  factor  1  should  be  retained  in  order  to 
make  the  terms  homogeneous,  giving 

(d 


in  which  1  represents  unit}'  of  volume. 

In  representing  these  quantities  geometrically,  we  will  for  conven- 
ience use  straight  lines,  a  method 
strictly  correct  at  a  state. 

Let  A  be  the  initial  state,  0  Vt 
—  0  Vi  =  1,  be  the  increase  of 
volume,  A  JSthe  isothermal  through 
A,  or  if  the  isothermal  be  a  curved 
line,  then  A  B  will  be  tangent  to 
the  isothermal,  A  <pt,  B  <pi.  adi- 
abatics  indefinitely  extended  ;  then 


FIG.  h. 
If  a  b  be  an  isothermal  one  degree  higher  than  A  B,  theu 


and 


Aa=(^\  , 


AabB  =  Aa  •  t>,  e,  =  (-£-      •  1. 


ADDENDA. 


443 


If  the  area  <p\  A  B  q*  be  cut  by  isothermals  CD,  E  F,  etc.,  differing 
by  unity,  then 

AabB  =  ABDC=  C  D  F  E,  =,  etc., 

and 

(d  H\  (d  p\ 

^AB^=(T-c)r=  rfo?X     L 


The  next  equation  on  page  132,  becomes 


dT\dv)r 


TM        •    1. 


In  this  case,  the  temperature  of  the  working  fluid  is  increased, 
be  the  increase  of  pressure  for  an  in- 
crease of  temperature  of  one  degree 
and  »t  V,  =  1.     Then 

l  •— —  1     =  <z>i  A  B  ©a :   and  the  left 


member  is  the  heat  necessary  to  be 
added  to  that  of  <p\  A  B  q>3  in  order 
to  increase  the  temperature  one  de- 
gree, and  maintain  it  at  that  tempera- 
ture while  it  works  under  the  pres- 
sure   z'j   a,   through   the  space  Vi  «a,       v      Td  ^ 
provided    the  pressure    is  increased                        FIG.  i. 
uniformly  with  increase  of  temperature.     The  right  member  represents 
the  difference  of  the  external  works  during  the  expansions  at  tempera- 
ture T  and  T  +  1  plus  the  difference  of  the  internal  works  during  the 
same  expansions.     We  have 


The  difference  of  the  internal  works  in  expanding  along  the  two 
isothermals,  A  B  and  a  b,  respectively,  equals  the  difference  of  the  in- 
ternal works  along  the  paths  A  a  and  B  b,  respectively,  and  the  latter  is 
as  given  on  page  119, 

K*.    —  -S~B    —    T  -j — -,  (z'a  —  Vi  =  1)  =  t  - — r    .  1. 
(IT1  d  T* 

The  next  equation  on  page  132  is 


444 


ADDENDA. 


This  represents  the  heat  absorbed  at  constant  volume  for  an  increase 
of  one  degree  of  temperature.  It  is,  as  shown  by  the  right  member,  the 
heat  which  makes  the  substance  hotter,  represented  by  C,  plus  that  which 
does  internal  work.  It  may  be  represented  on  a  diagram  of  energy  by 
the  area  tpi  A  B  <pt,  in  which  A  B  is  the  in- 
crease of  pressure  due  to  an  increase  unity  of 
temperature  and 


The  sum  of  these  two  quantities  is  the  specific 
heat  at  state  A.  The  heat  absorbed  in  raising  the 
temperature  one  degree  will  be,  making  ra  = 
r  +  1  in  equation  (106),  or 


The  last  term,  generally,  cannot  be  integrated  with  rand  r,  mutually 
dependent  variables  as  they  will  be,  except  for  isothermal  expansion.  The 
internal  work  will  be  the  same  whether  the  expansion  be  isothermal  or 
adiabatic,  if  indefinitely  extended  ;  for  the  two  paths  are  asymptotic  to 
each  other,  and  the  initial  states  are  the  same,  so  that  the  two  paths 
will  form  a  closed  cycle. 

Let  A  TI  and  B  r,  be  two  isother- 
mals  differing  by  unity  ;  then  may  the 
difference  of  the  internal  works  in 
expanding  along  these  isothermals  be 
represented  by  the  shaded  area  A  ami  T, 
and  this  area  will  equal  the  shaded  one 
in  figure.;',  or 

TI  A  a  mi  =  g>i,  A  a  n\. 

FIG.  k. 

Conceiving  the  expansion  to  be  isothermal,  r  will  be  constant  during 
the  ^-integration,  and  v  constant  for  the  r-integration.  If  the  equa- 

tion of  the  gas  be  p  v  =  R  r  -  ~t  then  for  an  increase  of  one  de- 
gree of  temperature 

* 


T  (r  +  1) 


Page  132.—  In  the  equation 


ADDENDA. 


445 


the  value  (— — j  is  the  heat  absorbed  for  an  increase  of  temperature 
of  one  degree  (or  strictly  it  is  the  rate  at  which  heat  is  absorbed  per 
degree  of  temperature)  at  constant  volume,  d  ( - —  \  is  an  element- 
ary increase  of  this  heat,  and  3—  ( —-  \  is  the  elementary  amount  for 
a  v  \  a  T  /  v 

an  increase  of  volume  equal  to  unity.  The  value  is  more  readily  seen 
from  the  right  member.  Referring  to  equation  (105),  page  119,  it  will 
be  seen  that  the  right  member  is  the  heat  absorbed  in  doing  internal 
work  for  an  expansion  unity  of  volume,  provided  it  is  uniform  through- 
out that  volume.  Therefore  r  ( -7-^  j  may  be  considered  as  the  ordinate 

at  any  point  whose  abscissa  is  ®  of  the  shaded  part  tpi,  A  a  HI,  or  of 
TI  A  a  mi,  the  latter  of  which  will  be  used  in  the  solution  of  problems 
since  r  will  then  be  constant. 
In  the  equation  (page  132) 


dH\    __T/dv 

dp)r  (J7 


the  left  member  is  the  heat  absorbed  dur- 
ing isothermal  expansion  fora  fall  of  pres- 
sure equal  to  unity.  Let  A  B  be  an  isother- 
mal, Bb  =  1,  then  will 


FIG.  I. 


Let  c  e  be  an  isothermal  one  degree  higher  than  A  B;  prolong  A  b  to 
d,  and  draw  B  e  parallel  to  A  d;  then 


Since  b  B  is  negative,  we  have 


Ad  •  bB=   j^     -  (— 


Page  134.—  The  differential  of  a  function  of  any  number  of  vari- 
ables may  be  found  by  well-known  rules,  the  result  being  an  exact  dif- 
ferential ;  but  it  is  often  difficult  to  find  the  primitive  of  a  differential. 
A  transformation  is  often  necessary  in  order  to  render  an  equation  or  an 


446  ADDENDA. 

expression  an  exact  differential.     Thus,  the  first  member  of  the  equa 
tion 

-r-  y*)  d  x  =  °> 


is  not  an  exact  differential  as  it  stands,  for  it  does  not  satisfy  the  condi- 


tiood(a?y)  =  diL+J^-;  but  multiplying  the  equation  by  2  a?  it  be 
d  x  d  y 

comes 

2  a*  y  d  y  +  2  (1  +  y»)  x  d  x  =  o, 

and  then 


_ 

dx  dy 

for  it  reduces  to 


and  the  integral  will  be 

a*  (1  +  y«)  =  e. 

If  the  expression  does  not  equal  zero  we  may  write  it 
d  <p  —  M  d  x  —  N  d  y  =  o, 

which  is  a  differential  equation  of  three  variables,  of  which  two  may  be 
independent.  According  to  the  theory  of  Differential  Equations,  the 
last  equation  can  be  integrated  if 

Mdx  +  Ndy  =  o 

can  be  integrated  (Boole's  Dif.  Eqs.,  4th  ed.,  page  276).  An  integrating 
factor  always  exists  for  the  latter  equation,  although  it  cannot  always 
be  found.  The  equation 


on  page  48  is  an  equation  of  three  variables,  of  which  T  and  v  are  inde- 
pendent of  each  other.     If 


is  integrable,  then  is  the  former  equation  also  integrable.    It  has  been 
found  by  some  writers  that  the  integrating  factor  is  —  ;  hence, 


ADDENDA. 


447 


is  integrable.     For  instance,  substituting  the  value  of  Kv  from  page  119, 
it  may  be  found  that 

(p*  —  <?!  =  C  log  r  -\-f  (0). 

The  function  of  v  may  be  found  in  some  cases  when  the  equation  of 
the  gas  is  known.  (Boole's  Dif.  Eqs.,  page  49.)  The  integral  is  of  little 
value  except  when  the  temperature  is  constant. 

If  a  substance  be  worked  in  a  cycle  the  resultant  internal  work  will  be 
zero,  and  the  resultant  internal  energy  will  be  constant  ;  hence  the 
expressions 

*J*.          d  E,  dll—  pdv, 

T 

page  135,  etc.,  are  exact  differentials. 

Page  142. — The  thermodynamic  function  of  a  perfect  gas  between 
finite  limits  becomes 

<p2  _  (PI  =  C  log  —  • 

Let  the  expansion  from  Vi  to  v*  be  iso- 
thermal along  A  C.  Let  ED  be  an  iso- 
thermal one  degree  lower  than  A  C, 
A  gti  C  <pa  adiabatics  ;  then 

A  CD  E  =  Rlog-- 

Equation  (36),  page  55,  makes  this  evi- 
dent.    Take  F  on  the  adiabatic  through     o      *i  v~2 
B,  one  degree  lower  than  the  temperature                    FIG-  m- 
of  B,  and  similarly  for  all  points  between  Fund  D  and  draw  F  D  ;  then 

Clog  ^  =  CBFD. 

Let  A  B  be  any  arbitrary  path  of  the  fluid  ;  between  A  and  B  draw 
consecutive  adiabatics,  e  d  and  /  c,  etc.,  and  make  efgk  =  abcd,  etc., 
thus  determining  a  point  g,  and  in  a  similar  manner  find  other  points  as 
p  o,  etc.,  and  draw  E  g  o  F;  then 

CBFD. 


Page  179.— The  theoretical  efficiency  for  each  of  the  three  cases 
considered  is  about  20  per  cent.  But  no  allowances  were  made  for 
clearance,  condensation,  nor  for  the  imperfection  of  the  indicator  diagram. 
In  order  to  reach  more  nearly  to  actual  conditions,  assume  that  clearance 


448  ADDENDA. 

reduces  the  efficiency  3  percent.  ;  condensation,  say  25  percent.  ;  imper. 
fection  of  the  diagram,  say,  4  per  cent.  Then  will  the  practical  effi- 
ciency be 

0.97  X  0.75  X  0.96  =  0.70  nearly 

of  the  theoretical  value  ;  hence  the  actual  would  be 

20  X  0.70  =  14  per  cent, 
which  has  been  exceeded  in  practice,  as  shown  on  page  221. 

Page  234.—  EXERCISE  2.  If  fa  of  the  heat  is  abstracted  by  the  re- 
frigerator, then  -ft  does  work,  and  the  efficiency,  neglecting  friction  and 
leakage,  will  be  30  per  cent. 

If  1670000  be  -ft,  then  will  the  entire  heat  absorbed  be  1670000  ~  0.3 
=  5566666  ft.  Ibs.  The  heat  removed  by  the  refrigerator  will  be  250  X 
18  X  778  =  3501000  ft.  Ibs.  ;  hence,  if  the  data  is  correct,  the  total 
heat  would  be  1670000  -f  3501000  =  5,171,000  ;  and  the  heat  unac- 
counted  for  would  be  5566666  —  5171000  =  395666  foot-pounds,  which  is 
about  7  per  cent,  of  the  whole.  The  data  are  inconsistent  unless  it  be 
assumed  that  about  7  per  cent  is  consumed  by  the  friction  of  the  engine. 

During1  a  test  of  a  De  La  Vergne  ice-making  machine  at  Memphis, 
Tenn.,  from  July  21st  to  August  10th,  1888,  1,221,172  pounds  of  com- 
mercial ice  were  produced  by  a  consumption  of  180  597  pounds  of 
bituminous  coal  ;  giving  an  average  for  the  20  days  of  6.76  pounds  of  ice 
per  pound  of  coal.  If  the  coal  cost  $4  per  2000  pounds,  then  would  the 
coal  bill  be  fa  of  a  cent  per  pound  of  ice. 

Latent  heat  of  vaporization  of  Ammonia,  NH,.—  Dr. 
Von  Strombeck  determined  this  value  at  32.45°  F.  and  Regnault  found 
the  same  at  53.01°  F.  (For  the  latter  see  a  paper  by  Professor  .l.-irolms 
in  the  Trans,  of  the  Am.  Soc.  Neck.  Eng.,  vol.  xii.)  These  results  fur- 
nish a  check  to  the  theoretical  formula  (360),  page  332.  The  following 
are  the  results  : 

At  32.45°  F.  the  author  finds  by  Table  VI  ..............  Ae   =  535.18 

At  32.45°  F.  Dr.  Von  Strombeck  finds  ..................  7<e   =  534.2 


Difference  .......................................  0.98 

or  about  £  of  one  per  cent. 
At  53°  F.  the  author  finds  .............................  7<e  =  522.39 

At53°F.  Regnault  exp.  gives  .........................  7<e=  521.64 

Difference  ........................  '.  ...............  0.75 

or  about  \  of  one  percent. 

These  results  indicate  that  Table  VI.  is  practically  exact. 


ADDENDA. 


449 


Latent  heat  of  the  vaporization  of  Sulphur  Dioxide,  SO,. 


Deg.  F. 

Experimental 
results. 

Formula  (c)  page 

Difference. 

—10 
0 
5.74 
10.50 
20. 

170.21 
165.06 
160.88 
157.18 
152.06 

173.68 
171.26 
169.92 
168.56 
165.59 

3.47 
6.20 
8.04 
11.38 
13.53 

From  these  results  it  will  be  seen  that  the  experimental  and  theoretical 
results  depart  more  and  more  with  increase  of  temperature.  The  fol- 
lowing formula  represents  the  results  of  these  experiments,  for  the  30° 
here  given,  with  sufficient  accuracy 

Ae   =  177.65  -  0.385  T, 

in  which  T  is  degrees  F.  At  140°  F.  this  gives  Ae  =  123.75,  while 
the  theoretical  table  gives  108.37,  a  difference  of  15.38  heat  units,  or  14 
per  cent.  It  is  probable,  therefore,  that  the  above  formula  gives  more 
accurate  results  than  formula  (c),  page  357  ;  and  this  would  diminish 
the  computed  volume  of  the  pound  of  vapor  in  Table  VII.  in  the  same 
ratio,  being  an  average  of  about  5  per  cent,  within  the  limits  for  which 
it  is  ordinarily  used  in  practice. 

The  equation  for  the  flow  of  steam  from  a  straight  uniform 
tube  of  large  diameter  into  a  straight  uniform  tube  of  small  diameter  is  : 


in  which  J  is  the  mechanical  equivalent  of  heat,  and  g  is  the  acceleration 
due  to  gravity  ;  Q  is  the  heat  given  to  the  steam  at  the  orifice  where  the 
small  tube  joins  the  large  one  ;  «»  is  the  velocity  in  the  large  tube  at  a  dis- 
tance from  the  orifice,  and  »b  the  velocity  in  the  small  tube  also  at  a  dis- 
tance from  the  orifice  ;  p*  and  pb  are  the  pressures  in  the  tubes  A  and  B, 
p&  being  the  larger,  and  h&  and  Ab  are  the  latent  heats  of  vaporization,  and 
qA  and  q\>  the  heats  of  the  liquid  corresponding  ;  x&  is  the  part  of  one 
unit  of  weight  of  the  fluid  in  the  tube  A  which  is  dry  steam,  and  1  —  #a  is 
the  part  which  is  water  mingled  with  the  steam  ;  *b  is  the  corresponding 
quantity  for  the  tube  B  ;  6  the  weight  of  unity  of  volume  of  the  liquid. 

It  is  assumed  that  neither  tube  gives  heat  to  the  steam  or  receives  heat 
from  them,  and  that  the  friction  of  the  fluid  on  the  sides  of  the  wall  can  be 
neglected.  The  heat  Q  is  supposed  to  be  given  at  the  orifice  ;  it  is  com- 
monly assumed  to  be  zero,  in  which  case  the  flow  is  said  to  be  adiabatic. 

The  value  of  .ra  must  be  determined  by  experiment.  o\>  can  then  be 
determined  by  the  equation  : 


450 
2 


REFRIGERATING   MACHINE. 


lllliili 


"5  MO  =  c  £  s? 

lil^fp 


\le9\ 


- 

h 


ill 

V5« 

:W"3 

iff 


!S 


:=      = 


! :  i  if  I  I  till 
=  i  j  !!  i  1 

c*  ^  If  !*=<§ 


I  |  j 

E      S 


I    J   J     P| 

S     ?     H     •<£ 


fc.  QJ  o-^^ 


KEFKIGERATING   MACHINE 


451 


^2a05*.    •      °-      O^c-^gSSj 

3«SSS3S  §    S5Srt0f ' 


-00      ro" 


SSSiSSS 


mi  i  si 

"SHI'S  go 


;siiii^js 


jwysiwh 


•  •  13  §  • 

!  i  i  I  : 


|^    |||    |||  |1  E|||  |'l  •  §  |  *  B  iJl^l 


*i*if!!  i  f  ii^s^ii.- 111,  ii 

-slilllll  lilllSl'8*  5*'  5sgg 
IlilSlS  I  limit*?*  I§1  isij 

-  - 


452 


REFRIGERATING   MACHINE. 


TEST  OF  A  PULSOMETEK  BY  C.  G.  ATWATER   AND   CHARLES  B.  HODGES, 

OP  THE  CLASS  OF  '91,  STEVENS  INSTITUTE,  under  the  supervision 
of  the  Department  of  Tests.  The  pump  was  taken  from  the  ordi- 
nary stock  of  machines  on  hand,  and  was  known  as  No.  6.  Tests 
were  for  three  hours  each. 


or  THE  TEST. 


DATA  AND  RESULTS. 

1 

2 

3 

4 

Strokes  per  minute        
Bteam  pressure  in  pipe  before  throttling,  Ibs. 
"           ••        "     '•     after 

71. 
114. 
19. 

60. 
110. 
30. 

57. 
127. 
43.8 

64. 
104.3 
2C.1 

'      temperature  uf  ter  throttling,  Deg.  F.  . 

270.4 

277.4 

309. 

850.1 

'      amount  of  supei  heating.  Deg.  F  
"      total  heat  of,  above  highest  tempera- 
ture of  water,  B.  T.  U  

3.1 
1118.67 

3. 
11U.44 

17.4 
1187. 

1.4 
1121.2 

'     used  as  determined  from  temp.,  Ibs.  .. 

1617. 

921 

1518. 

1019.9 

^Tater  pumped  Ibs 

404786 

180362 

•,'•>  IN) 

248063 

"     temp.,  before  entering  pump,  beg.  F. 
''     temperature  after  leaving  pump,Deg.F. 
44          rise  of    Deff  F 

75.15 
79.62 
4  47 

80.6 
86.1 
5  5 

76.3 
83.79 
7  49 

70.25 
74.8 
4  55 

44     heat  absorbed,  B.  T.  U  

1815127. 

1024993. 

1704498. 

IHiMT. 

11     head  by  gauge  on  lift,  f  t  .        

29.90 

M.0i 

54.05 

29.9 

44              4*      **       4t  suction   * 

12  26 

12  2J 

19  67 

19  67 

41              l4      "       total  (H)  

42.16 

66.31 

73.72 

4957 

44  measure  on  lift.     

253 

50.3 

503 

25.30 

14              44        44        44  suction 

7  ft 

16  3 

1630 

"               "         '4          total  (h)  

828 

57.8 

66.6 

41.60 

Efficiency   of   pump   compared    with    total 
work,  (h)  •+•  (H)  

0.777 

0877 

0.911 

0.839 

Total  work  a»  per  gauges,  ft.  Ibs  
Efficiency  of  the  pulsometer  
44           "  plant  exclusive  of  boiler  

UJM08W. 

0.012 
0.0093 

12335940. 
0.0155 
0.0186 

167951112. 
0.0126 
0.0115 

lenaeto. 

0.0138 
0.0116 

44          "      "     if  that  of  boiler  and  fur- 

nace be  0.7  

0.0065 

0.0095 

0.0080 

0.0081 

Duty  of  pump  per  100  Ibs.  coal  if  1  Ib.  evap- 
orates 8  Ibs.  water  

8409C20. 

10712800. 

8847200. 

%*KMO. 

Duty  If  1  Ib.  coal  evaporates  10  Ibs.  water.  .  .  . 

1051  1400. 

I3S9100.) 

110U8500. 

12036:100. 

Of  the  two  tests  having  the  highest  lift  (54.05  ft.),  that  was  more  effi- 
cient which  had  the  smaller  suction  (12.26  ft.),  and  this  also  was  the  most 
efficient  of  all  the  tests.  But  in  the  other  two  tests  having  the  same 
lift  (29.9),  that  was  most  efficient  which  had  the  greater  suction  (19.67). 
The  pressures  used,  19,  30,  43.8,  26.1,  were  made  to  follow  the  order  of 
total  heads,  but  are  not  proportional  thereto.  They  doubtless  have 
much  to  do  with  the  efficiency,  but  no  attempt  was  made  to  determine 
the  pressure  which  would  give  the  highest  efficiency.  The  first  tost 
compared  with  the  other  three  is  somewhat  paradoxical.  Thus,  it  is 
peculiar  that  a  pressure  of  19  pounds  of  steam  should  produce  a  greater 
number  of  strokes  and  pump  over  50  per  cent,  more  water  than  26.1 
pounds  of  steam,  the  lift  being  the  same. 


.REFRIGERATING   MACHINE.  453 

TEST  OP  THE  NEW  YORK  HYEIA  ICE-MAKING  PLANT,  BY  A.  G. 
HUPFEL,  H.  E.  GKISWOLD,  AND  WILLIAM  P.  MACKENZIE,  FOR 
GRADUATING  THESIS,  1893.  under  the  supervision  of  the  Department 
of  Tests  of  Stevens  Institute  : 

Net  ice  made  per  pound  of  coal  in  pounds 7. 12 

Pounds  of  net  ice  per  hour  per  horse-power 37.8 

Net  ice  manufactured  per  day  (12  hours)  in  tons 97 

Average  pressure  of  ammonia  gas  at  condenser  in  pounds  per 

square  inch  above  the  atmosphere ...;...       135.2 

Average  back  pressure  of  ammonia  gas  in  pounds  per  square 

inch  above  the  atmosphere 15.8 

Average  temperature  of  brine  in  freezing  tanks  in  degrees 

Falir 19.7 

Total  number  of  cans  filled  per  week 4,389 

Ratio  of  cooling  surface  of  coils  in  brine  tank  to  can  surface.  7  to  10 

Ratio  of  brine  in  tanks  to  water  in  cans 1  to  1 .2 

Ratio  of  circulating  water  at  condensers  to  distilled  water. .  .26  to  1 

Pounds  of  water  evaporated  at  boilers  per  pound  of  coal 8 . 085 

Total  horse-power  developed  by  compressor  engines 444 

Percentage  of  ice  lost  in  removing  from  cans 2.2 

APPROXIMATE  DIVISION   OF   STEAM  IN  PER   CENTS.   OF  TOTAL  AMOUNT. 

Compressor  engines 60.1 

Live  steam  admitted  directly  to  condensers 19.7 

Steam  for  pumps,  agitator  and  elevator  engines 7.6 

Live  steam  for  reboiling  distilled  water 6.5 

Steam  for  blowers  furnishing  draught  at  boilers 5.6 

Sprinklers  for  removing  ice  from  cans 0.5 

EXERCISES. 

1.  Required  the  specific  heat  of  air  at  500°  F.  absolute,  the  path  being 
p  =  10  v  (p,  pounds  per  square  foot ;  «,  volume  of  a  pound  in  cubic 
feet).     The  specific  heat  is  the  heat  absorbed  in  raising  one  pound  one 
degree. 

2.  If  the  equation  of  a  superheated  vapor  be  p  v  =  R  r  —  Cp*  (see 
p.  414)  ;    required  the  heat  absorbed  at  the  constant  pressure  pi  in 
expanding  from  vl  to  tv 

3.  If  the  equation  of  the  gas  be  p  v  =  R  r  —  —  and  of  the  path  of 

the  lluid,  p  =  m  v  -\-  n  ;  required  the  heat  absorbed  in  expanding  from 
state  vl  =  12,  pt  =  4000  to  state  n,  =  24,  p«  =  5000. 

4.  Find  the  internal  and  the  external  work  of  expanding  the  gas 


454  REFRIGERATING    MACHINE. 

pv  =  Rr —t  at  the  constant  pressure  pi  from  vl  to  ?2,  and  leave  the 

final  result  without  r, 

5:  Find  the  thermodynamic  function  for  the  gas  whose  equation  is 

p  v  =  R  T  —   — s  for  isothermal  expansion. 

6.  Find  the  velocity  of  discharge  of  a  perfect  gas  from  an  orifice, 
the  temperature  remaining  constant.     Also  the  weight  per  second. 

7.  Write  a  formula  for  the  pressure  of  a  saturated  vapor  in  terms 
of  the  volume  of  a  pound. 

8.  Required  the  difference  in  the  elevation  of  two  stations  at  which 
water  boils  at  atmospheric  pressure  respectively  at  212°  F.  and  180°  F. 

9.  A  vessel  containing  two  cubic  feet  of  fluid,  one-fourth  of  which 
by  weight  is  steam  and  the  remainder  water  ;  required  the  work  neces- 
sary to  compress  the  vapor  to  water — in  one  case  adiabiatically,  and  in 
another  isotliermally. 

10.  A  frictiouless  piston,  in  an  upright  cylinder,  rests  on  a  pound  of 
water.     Heat  is  absorbed  under  a  pressure  of  6  atmospheres  absolute 
until   the  piston   has  swept  over  two  cubic  feet,  then  expansion   is 
adiabatic  until  the  pressure  is  reduced  to  two  atmospheres  absolute,  then 
compressed  isothermally  until  by  adinbatic  compression  the  vapor  will 
be  reduced  to  water  at  the  initial  pressure.     Required  (1)  the  heat  ab- 
sorbed ;  (2)  the  clearance  ;  (3)  the  entire  stroke  of  the  piston  ;  (4)  the  heat 
emitted  during  compression  ;   (5)  the  work  done  in  the  cycle  ;   ((>)  t In- 
efficiency of  the  cycle. 

11.  Find  difference  of  external  works  in  expanding  a  gas  adiabati- 
cally  and  isothermally  from  0,  to  2  r,. 

12.  Find  heat  absorbed  by  a  liquid  in  raising  the  temperature  from 
40°  to  200°,  if  specific  heat  c  =  1  -f  a  TJ.      If  c  =  1  -  a  (T  -  T0).     If 
c=  1  -a(T-T#. 

13.  What  is  the  specific  heat  of  a  gas  at  constant  temperature  ? 

14.  Find  the  thermodynamic  function  for  air  from  state  TI,  pt  to 
Tt,  p-t,  independently  of  v. 

15.  A  single-acting  engine  (vertical)  has  one  pound  of  fluid  (water)  at 
the  lower  end,  on  which  rests  a  frictiouless  piston.     By  the  absorption 
of  heat  the  piston  is  raised  against  an  absolute  pressure  of  9  atmospheres 
until  the  volume  swept  over  by  the  piston  is  twice  the  volume  of  the 
dry  saturated  vapor  at  that  pressure,  then  it  expands  adiabatically  as  a 
perfect  gas  until  the  pressure  is  reduced  to  3  atmospheres,  then  re- 
frigerated at  constant  volume  until  the  pressure  is  reduced  to  2   atmos- 
pheres, then  refrigerated  at  constant  pressure  until  the  volume  is  such 
that  when  compressed  adiabatically  it  wrill  be  reduced  to  a  liquid  at 
7  atmospheres,  and  its  temperature  then   raised  under  a  pressure  of 
9  atmospheres  to  the  initial  state. 


KEFKIGEEATIJSG   MACHINE.  455 

Find: 

1.  Indicator  diagram. 

2.  Volume  at  the  beginning  of  adiabatic  expansion. 

3.  Degrees  of  superheating. 

4.  Volume  swept  over  by  the  piston. 

5.  Ratio  of  expansion. 

6.  Temperature  at  end  of  expansion. 

7.  Temperature  at  beginning  of  back  stroke. 

8.  Degrees  of  superheating  on  back  stroke. 

9.  Volume  at  end  of  refrigeration,  so  that  when  compressed  adia- 
batically  it  will  all    be  reduced  to  liquid  at  7  atmospheres. 

10.  Temperature  at  end  of   back  stroke  when  fluid  is   reduced  to 
water. 

11.  Clearance. 

12.  Work  done  in  the  cycle. 

13.  Mean  effective  pressure. 

14.  Mean  forward  pressure. 

15.  Heat  absorbed  in  the  cycle. 

16.  Heat  emitted. 

17.  Pounds  of  coal  necessary  to  supply  the  heat  for  100  *ycles,  if 
each  pound  contains  14,500  heat  units. 

18.  Pounds  of  water  necessary  to   produce  the  refrigeration  for  100 
cycles— temperature  ranging  from  60°  to  80°. 

19.  If  length  of  stroke  equals  diameter  of  cylinder,  find  diameter  of 
cylinder 

20.  The  pounds  of  water  necessary  to  develop  a  horse  power  if  piston 
speed  be  200  feet  per  minute. 

21.  The  pounds  of  coal  necessary  to  produce  the  horse  power  if  a 
pound  of  coal  has  1200  heat  units  and  the  efficiency  of  the  furnace  be 
0.70. 

16.  A  variety  of  exercises  may  be  made  similar  to  the  preceding.    We 
suggest  the  following  :  Let  the  volume  of  the  vapor  be  the  fraction  of  a 
pound.     Let  the  expansion  be  entirely  with  saturated  vapor  and  adia- 
batic.    Let  it  be  the  curve  of  saturation,  or  a  straight  line  entirely 
within  the  curve  of  saturation,  or  entirely  without  (superheated) ;  or  a 
line  crossing  the  curve  of  saturation.     Let  the  indicator  card  be  entirely 
within  the  curve  of  saturation  and  bounded  by  any  assumed  lines. 

17.  A  pound  of  water  at  60°  F.  is  within  a  closed  vessel  containing 
two  cubic  feet ;  required  the  heat  absorbed  by  the  fluid  in  raising  its 
temperature  335°  F. 

From  pp.  146  and  147  : 

d  H=  C 


456  REFRIGERATING   MACHINE. 

Let 

v  =  volume  of  mixture  of  liquid  and  vapor  =  2  cubic  feet. 
fa  =  specific  volume  of  1  pound  vapor  (see  table). 
t>i  =  specific  volume  of  liquid  =  0.017  nearly  for  water. 

Then, 

rl£ 

x  =  t-L^=  dr 

• 


a  —  b  T 
He 


(Eq.  (84));  II.  -  a  —  b  r, 


r*i 

d  H,        77,  d  \T  (dlr)) 


Reduce  and  find, 

A  n*  I        d  n 

•d  r 


/ra 
dTl 


H= 

'   dri 

18.  If  gas  flows  from  one  vessel  into  another  through  a  short  pipe, 
what  must  be  the  cross-section  of  the  pipe  that  q  pounds  will  flow  in  t 
seconds  at  constant  temperature  ? 

Let  Q  and  Q  be  the  volumes  of  the  vessels,  k  the  section,  P'  and  p' 
the  initial  pressure,  P'  >  p',  P&udp  the  pressures  at  time  t,  /i  the  co- 
*    efficient  of  velocity,  Fthe  velocity  at  the  time  t,  and  w  the  weight  per 
cubic  foot. 
Then, 

V  = 


REFRIGERATING  MACHINE.  457 

These  give 


which  integrated  between  0  and  t  and  o  and  q  will  give  an  equation 
from  which  k  may  be  found. 

19.  Find  the  path  of  a  fluid  considered  as  a  perfect  gas  when  the  heat 
absorbed  xs  n  times  the  work  done. 

Make  npdv  =  d  If  in  equations  in  p.  50,  and  find 

Y  -  n  (y  -  1) 
p  v  =  B  (a  constant). 

Discuss  making  n  =  1,  2,  5-9,  oo .  What  value  of  n  makes  the  path, 
a  right  line? 

Specific  Heat  of  Aqua  Ammonia.— The  mean  of  six  deter- 
minations by  Ludeking  and  Starr  gives  0.886  (Am.  Jour.  Arts  and  Sc.). 
The  value  found  by  Hans  von  Strombuck  was  1.22876  (page  337), 
which  is  nearly  50  per  cent,  larger  than  the  above  value.  The  value 
found  by  theory  is  nearly  the  mean  of  the  two.  The  above  experi- 
menters inform  the  author  that  they  are  not  aware  of  any  error  in 
their  own  work,  neither  do  either  know  of  an  error  in  the  work  of 
the  other.  This  leaves  the  correct  value  in  doubt,  and  one  may  con- 
sider it  as  unity  until  determined  by  further  experiments. 


458 


LOGARITHMS    OF    NUMBERS. 


N 

01231 

56789 

D 

10 

11 

12 
13 
14 

0000      043      086      128      170 
414      453      492      531      569 
792      828      864       899      934 
1139      173      206      239      271 
461       492      523      553      584 

212  253  294  334  374 
607  645  682  719  755 
969  1004  1038  1072  1106 
303  335  367  399  430 
614  644  673  703  732 

42 
38 
35 
32 
30 

15 

i<; 

17 
18 
19 

17G1      790      818      847      875 
2041      068      095      122      148 
304      330      355      380      405 
553      577      601      625      648 

788      810      833      856      878 

903  931  959  9S7  2014 
175  201  227  253  279 
430  455  480  504  529 
672  695  718  742  765 
900  923  945  967  989 

28 
26 
25 
24 
22 

20 

21 
22 
23 
24 

3010      032      054      075       096 
222      243      263      284      304 
424      444      464      483      502 
617      636      655      674      692 

802      820      838      856      874 

118  139  160  181  201 
324  345  365  385  404 
522  541  560  579  598 
711  729  747  766  784 
892  909  927  945  962 

21 

20 
19 
19 

18 

25 

26 
27 

28 
29 

3979        997        4014        4031        4048 
4150       166       183      200      216 
314      330      346      362      378 
472      487      502      518      533 
624      639      654      669      683 

4065  4082  4099  4116  4133 
232  249  265  281  298 
393  409  425  440  456 
548  5(54  579  594  609 

698  713  728  742  757 

17 
16 
16 
15 
15 

30 

31 
32 
33 
34 

4771       786      800      814      829 
914      928      942      955      «j69 
5051      065      079      092      105 
185      198      211      224      237 
315      328       340      353      366 

843  857  871  886  900 
983  997  6011  5024  5038 
119  132  145  159  172 
250  263  276  289  302 
378  391  403  416  428 

14 
14 
13 
13 
13 

35 

36 
37 
38 
39 

5441       453       465       478      490 
563      575      587      599      611 
682      694      705      717      729 
798      809      821      832-      843 
911      922      933      944      955 

502  514  527  53'J  551 
623  635  647  658  670 
740  752  763  775  786 
855  866  877  888  899 
966  977  988  999  coio 

12 

13 
12 
11 
11 

40 

41 
42 
43 
44 

6021      031      042      053      064 
128      138      149      160      170 
232      243      253      263      274 
335      345      355      365      375 
435      444      454      464      474 

075  083  096  107  117 
180  191  201  212  222 
284  294  304  314  325 
385  395  405  415  425 
484  493  503  513  522 

11 
10 
10 
10 
10 

45 

46 
47 
48 
49 

6532       542       551       561       571 
623      637      646      656      665 
721      730      739      749      758 

812      821      830      839      848 
902       911       920      9:28      937 

580  590  599  609  618 
675  684  693  702  712 
767  776  785  794  803 
857  866  875  884  893 
946  955  964  972  981 

10 

50 

51 
52 
53 
54 

0990       098       7007       T016       7024 
7076      084      093      101       110 
160      168      177      185      193 
243      251      259      267      275 
324      332      340      348      356 

7033  7042  7050  7059  70(57 
118  126  135  143  152 
202  210  218  226  235 
284  292  300  308  316 
364  372  380  388  396 

8 
8 
8 

LOGARITHMS    OF    NUAIBEKS. 


459 


N 

01234 

56789 

D 

55 

56 
57 
58 
59 

7404      412     -419      427      435 
482      490      497      505      513 
559      566      574      582      589 
634      642      649      657      664 
709      716      723      731      738 

443     451     459     466     474 
520     528     536     543    551 
597    604    612    619     627 
672     679    686    694    701 
745     752    760    767    774 

8 
8 
8 

7 

7 

60 

61 
62 
63 
64 

7782      789      796      803      810 
853      860      868      875      882 
924      931      938      945      952 
993       8000       8007       8014       8021 
8062      069      075      082      089 

818    8^5    832     839    840 
889    896    90.^    910    917 
95.)     966    973    980    987 
8028     8035     8041      8048     8055 
096    102     109     116     122 

7 

65 

66 
67 
68 
69 

8129      136      142      149      156 
195      202      209      215      222 
261      267      274      280      287 
325      331      338      344      351 
388      395      401      407      414 

162     169     176     182     189 
228    235     241     248    254 
293    299    306    312     319 
357    363    370    376    382 
420    426    432    439    445 

7 
7 
6 
6 
6 

70 

71 
72 
73 
74 

8451      457      463      470      476 
513      519      525      531      537 
573      579      585      591      597 
633      639      645      651      657 
692      698      704      710      716 

482    488    494    500    506 
543    549    555    561     567 
603    609    615    621     627 
663    669    675    681     686 
722    727    733    739     745 

6 
6 
6 
6 
6 

75 

76 

77 
78 
79 

8751       756      762      768      774 
808      814      820      825      831 
865      871      876      882      887 
921      927      932      938      943 
976      982      987      993      998 

779     785     791    797     802 
837    842    848    854    859 
893    899    904    910    915 
949    954    960    965    971 
9004     9009     9015     9020     9025 

6 
6 
6 
0 
5 

80 
81 
82 
83 
84 

9031      036      042      047      053 
085      090      096      101       106 
138      143      149      154      159 
191       196      201      206      212 
243      248      253      258      263 

058    063    069     074    079 
112     117     122     128     133 
165     170     175     180     186 
217    222    227    232    238 

269    274    279    284    289 

5 
5 
5 
5 
5 

85 
86 
87 
88 
89 

9294      299      304      309      315 
345      350      355      360      365 
395      400      405      410      415 
445      450      455      460      465 
494      499      504      509      513 

320    325     330    335     340 
370    375    380    385    390 
420    425    430    435    440 
469    474    479    484    489 
518    523    528    533    538 

5 
5 
5 
5 
5 

90 

91 
92 
93 
94 

9542      547      552      557      562 
590      595      600      605      609 
638      643      647      652      657 
685      689      694      699      703 
731      736      741       745      750 

566     571     576     581     586 
614    619     624    628    633 
661    666    671     675     680 
708    713    717    722    727 
754    759     763     768    773 

5 
5 
5 
5 
5 

95 

96 
97 

98 
99 

9777      782      786      791      795 
823      827      832      836      841 
868      872      877      881      886 
912      917      921      926      930 
956      961      965      969      974 

800    805    809    814    818 
845     850    854    859    863 
890    894    899    903    908 
934    939    943    948    952 
978    983    987    991     996 

5 
5 
4 
4 
4 

460 


LOGARITHMS    OF    NUMBERS. 

PROPORTIONAL  PARTS. 


D 

1 

2 

3 

4 

5 

6 

•  7 

8 

9 

42 
38 
35 
32 
30 

4.2 
3.8 
35 
3.2 
3.0 

8.4 
7.6 
7.0 
6.4 
6.0 

12.6 
11.4 
10.5 
9.6 

9.0 

16.8 
15.2 
14.0 
12.8 
12.0 

21.0 
19.0 
17.5 
16.0 
15.0 

25.2 

22.8 
21.0- 
19.2 

18.0 

29.4 
26.6 
24.5 
22.4 
21.0 

33.6 
30.4 
28.0 
25.6 

24.0 

37.8 
34.2 
31.5 

28.8 
27.0 

28 
26 
25 
24 
22 

2.8 
2.6 
2.5 
2.4 
2.2 

5.6 
5.2 
5.0 
4.8 
4.4 

8.4 
7.8 
7.5 
7.2 
(J.O 

11.2 
10.4 
10.0 
9.6 

14.0 
13.0 
12.5 
12.0 
11.0 

16.8 
15.6 
15.0 
14.4 
13.2 

rj.o 

18.2 
17.5 
16.8 
15.4 

22.4 
20.8 
20.0 
19.2 
17.6 

25.2 
23.4 
22.5 
21.  G 
19.8 

21 

20 
19 

18 
17 

2.1 
2.0 
1.9 

.8 

.7 

4.2 
4.0 
8.8 
3.6 

3.4 

6.3 
6.0 
5.7 
5.4 
5.1 

8.4 
8.0 
7.6 
7.2 
6.8 

10.5 
10.0 
9.5 
9.0 

8.5 

12.0 
12.0 
11.4 
10.8 
10.2 

14.7 
14.0 
13.3 
12.6 
11.9 

10.8 
16.0 
15.2 
14.4 
13.6 

18.9 
18.0 
17.1 
16.2 
15.3 

16 

15 
14 
13 
12 

.0 
.5 
A 
.3 

.2 

o.2 
3.0 
2.8 
2.6 
2.4 

4.8 
4.5 
4.2 
3.9 
3.6 

0.4 
6.0 
5.6 
5.2 

4.8 

8.0 
7.5 
7.0 
6.5 
6.0 

y.G 

9.0 

8.4 
7.8 
7.2 

11.2 
10.5 
9.8 
9.1 
8.4 

12.8 
12.0 
11.2 
10.4 
9.6 

14.4 
13.5 
12.6 
11.7 

10.8 

11 

10 
9 
8 

7 

1.1 
1.0 

.9 

.8 

2.2 
2.0 
1.8 
1.6 
1.4 

3.3 
3.0 
3.7 
2.4 
2.1 

4.4 
4.0 
3.6 
3.2 

2.8 

5.5 
5.0 
4.5 
4.0 
3.5 

6.6 
6.0 
5.4 
4.8 
4.2 

7.7 
7.0 
6.3 
5.6 
4.9 

8.8 
8.0 
7.2 
6.4 
5.6 

y.9 

9.0 
8.1 
7.2 
6.3 

6 
5 

3 
2 

.6 
.5 
.4 
.3 
.2 

1.2 
1.0 
.8 
.6 
.4 

1.8 
1.5 
1.2 
.9 
.6 

2A 
2.0 
1.6 
1.2 

.8 

3.0 
2.5 
2.0 
1.5 
1.0 

3.G 
3.0 
2.4 
1.8 

1.2 

4.2 
3.5 
2.8 
2.1 
1.4 

4.* 
4.0 
3.2 
2.4 
1.6 

5.4 
4.5 
3.6 
2.7 

1.8 

Modulus  of  the  common  system  =  0.4342945. 

1*10 -MOWS  =  53^ 

e  =  2.71828128 
logl<te  =  0.4342945. 
log.N  =  2.302585  logloN. 


REDUCTION   TABLES.  461 

TABLE   II. 

REDUCTION  TABLES. 

CONVERSION  OF  ENGLISH  AND  METRIC  UNITS. 

1  Foot  =  0.3048  metre. 

Litre  (vol.  of  1  kilog.  water)  =  0.2202  gal. 

Gallon  (vol.  of  10  Ibs.  water)  =  4.541  litres. 

Kilogramme  per  sq.  mm.  =  1422.28  Ibs.  per  sq.  in. 

Lb.  per  sq.  inch  =  703.0958  kilog.  per  sq.  metre. 

Grain  =  0.0648  gram. 

Foot-pound  =  0.1383  metre-  kilog. 

1  Atmosphere  =  14.7  Ibs.  per  sq.  in.  =  10334  kilog.  per  sq.  metre  = 
29.922  inches,  or  760  mm.  of  mercury  =  33.9  ft.,  or  10£  metres  of 
water. 

1  Pound  av.  =  0.4536  kilog. 

1  Calorie  (kilog.  water  raised  1°  C.)  =  424  metre-kilog.  =  3.9683  B.  T.U. 
1  Eng.  heat  unit  (Ib.  water  raised  1°  F.)  =  778  ft. -Ibs.  =  0.254  calorie. 

FRENCH  MEASURES  IN  EQUIVALENT  ENGLISH  MEASURES. 
MEASURES  OP  LENGTH. 

1  Millimetre  =  0.03937079  inch,  or  about',1?  inch. 
1  Centimetre  =  0.3937079  inch,  or  about  0.4  inch. 
1  Decimetre  =  3.937079  inches. 
1  Metre  =  39.37079  inches  =  3.28  feet  nearly. 

1  Kilometre   =  39370.79  inches. 

MEASURES   OF   AREA. 

1  sq.  decimetre  =  15.5006  sq.  inches. 

1  sq.  metre          =  1550.06  sq.  inches,  or  10.764  sq.  ft. 

MEASURES  OF  WEIGHT. 

1  Gramme         =  15.432349  grains. 

1  Kilogramme  =  15432.349  grains,  2.2046  Ibs.,  or  2.2  Ibs.  nearly. 

TWO  UNITS  INVOLVED. 

1  Gramme  per  sq.  centimetre        =  2.048098  Ibs.  per  sq.  foot. 

1  Kilogramme  per  sq.  metre          =0.2048098      " 

1  Kilogramme  per  sq.  millimetre  =  2.048098 

1  Kilogramme  metre  =  7.23314  ft. -Ibs. 

=  7i  ft.  Ibs.  nearly. 

1   force  de  cheval  =  75  kilogrammetres  per  second,    or  542*  foot- 
pounds per  second  nearly.     1  horse-power  =  550  foot  pounds  per  second. 


462 


TABLES. 


TABLE   III. 


LIQUIDS  AND  SOLIDS. 


Weight  of  a  cu- 
bic foot  of  the 
substance. 

Specific  Grav- 
ity. 

Expansion. 
-Liquids 
per   unit   of 
volume,    So- 
lids per  unit 
of       length, 
from   a*>  to 
212°  F. 

Specific 
Heat. 

Water,  pure  (at  39M  F.). 
"      sea,  ordinary  
Alcohol    pure       

w. 

62.425 
64.05 
49  38 

8.  G. 

1.000 
1.026 
0  791 

E. 
0.04775 
0.05 
0  1112 

C. 

1.000 

0622 

"        proof  spirit  
Ether    

57.18 
44  70 

0.916 
0  716 

0  517 

Mercury 

848  75 

13.596 

0.018153 

0033 

Naphtha       

52  94 

II  MS 

0  434 

Oil    linseed     

58  68 

0  940 

0  08 

"    olive     

57.12 

0  915 

0.08 

'  '    whale       

57  62 

0.923 

"    of  turpentine  
Petroleum  

54.31 
54  81 

0.870 
0  878 

0.07 

.434 

Ice            

57  5 

0  92 

.504 

Brass          

487  to  533 

7.8  to  8  5 

.00216 

Bronze   

524 

8.4 

.00181 

Copper  

537  to  556 

8  6  to  8.9 

.00184 

0951 

Gold 

1186  to  1224 

19  to  19  6 

0015 

0298 

Iron    cast  

444 

7.11 

.0011 

480 

7  69 

0012 

1138 

Lead 

712 

11  4 

0029 

0293 

1311  to  1373 

21  to  22 

0009 

0314 

Silver 

655 

10  5 

002 

0557 

Steel 

490 

785 

0012 

Tin 

462 

74 

0022 

0514 

Zinc 

436 

7  2 

00294 

0927 

PROPERTIES   OF   GASES   CONSIDERED   PERFECT.    463 

TABLE   IY. 

PROPERTIES  OP  GASES  CONSIDERED  PERFECT. 


I. 

NAME  op  THE  GAS. 

II. 

Chemical 
Compo- 
sition. 

III. 
Density. 

IV.             V. 

Specific  Heat  at 
Constant  Pressure 

VI           VII. 
Specific  Heat  j.t 
C'nstantVohm  e 

compared     corn- 
Weight        pared 
for          Volume 
Weight     for  Vol- 
with       umewith 
Water.          Air. 

com- 
pared 
Weight 
for 
Weight 

wia 

Water. 

com- 
pared 
Volume 
for 
Vo'.mr.e 
with 
Air. 

Atmospheric  Air  
O  xyi;en  

0, 

Na 

1 
1.1056 
0.9713 

0.0692 
2.4502 
5.4772 

1.0384 
0.9673 
1.2596 

1.5201 
1.5241 
0.6219 

2.2113 
1.1747 
2.6258 

0.5527 
4.1244 
0.9672 

0.5894 
2.6942 
4.6978 

1.1055 
1.5890 
2.5573 

3.1101 
2.2269 
3.7058 

3.4174 
2.0036 
3.0400 

5.8833 
4.7464 

6.2667 
6.6402 

8.9654 

0.2375 
0.21751 
0.24380 

3.40900 
0  12099 
0.05552 

0.2317 
0.2450 
0.1852 

0.2169 
0.2262 
0.4805 

0.1544 
0.2432 
0.1569 

0.5929 
0.1567 
0.4040 

0.5084 
0.3754 
0.5061 

0.4580 
0.4534 
0.4797 

0.4008 
(1.2738 
0.1896 

0.2293 
0.4125 
0.4008 

0.1322 
0.1347 

0.1122 
0.1290 

0.0939 

1 
1.013 
0.997 

0.993 
1.248 
1.280 

1.013 
0.998 
0.982 

1.39 
1.45 
1.26 

1.44 
1.20 
1.74 

1.38 
2.72 
1.75 

1.26 
4.26 
10.01 

2.13 
3.03 
5.16 

5.25 
2.57 
2.96 

3.30 

3.48 
5.13 

3.27 
2.69 

2.96 
3.61 

3.54 

0.1689 
0.1551 
0.1727 

2.411 
0.0928 

0.0429 

0.1652 
0.1736 
0.1304 

0.172 
0.181 
0.370 

0.123 
0.184 
0.131 

0.468 
0.140 
0.359 

0.393 
0.350 
0.491 

0.395 
0.410 
0.453 

0.379 
0.243 
0.171 

0.209 
0.378 
0.378 

0.120 
0.120 

0.101 
0.119 

0.086 

1 
1.018 

0.996 

0.990 
1.350 
1.395 

1  018 
0.997 
0.975 

1.55 
1.64 
1.36 

1.62 
1.29 
2.04 

1.54 
3.43 
2.06 

1.37 
5  60 
13.71 

2.60 

3.87 
6.87 

6.99 
3  21 
3.76 

4.24 
4.50 
6.82 

4  21 
3.39 

3.77 

4.67 

4.59 

Nitrogen  

Hydrogen 

Ho 
01, 
Bra 

NO 
C  O 
HC1 

CO, 
N2O 
H2  O 

SO, 
HaS 

cs» 

C  H4 
C  H  C13 
C2H4 

NH, 

C«H6 
C,0  Hi. 

CH4  O 
C  H60 
C   H100 

C   H10  S 

Chlorine 

Bromine  

Nitric  Oxide  
Carbonic  Oxide  
Hydrochloric  Acid... 

Carbonic  Acid  
Nitric  Acid  
Steam  

Sulphuric  Acid  
Hydro-sulphuric  Acid 
Carbonic  Di-sulphide. 

Carburetted  Hydrogen 
Chloroform  
Olefiant  Gas  

Ammonia  Gas  
Benzine  

Oil  of  Turpentine  
Wood  Spirit  

Alcohol  

Ether  

Ethyl  Sulphide 

Ethyl  Chloride 

C   H5C1 
C   H5Br 

C   H.C1S 
C   HB0 
C   H80a 

SiCl3 

PC13 

As  C13 
TiCl4 

SnCU 

Ethyl  Bromide  

Dutch  Liquid 

Aceton 

Butyric  Acid 

Tri  chloride  of  Silicon 
Tri-chloride  of  Phos-  j 
phorus     ...       )' 

Tri-chloride  of  Arsenic 
Tetra-chloride  of  Ti-  ) 
tanium     .     ...      j 

Tetra-chloride  of  Tin  . 

•i04 


TABLES. 


TABLE  Y. 
SATURATED  STEAM. 
p,  Pressure  per  square  inch. 
T,   Temperature  degrees  F.,  Eq.  (81),  page  97. 

v.  Volume  of   a  pound  of  saturated  steam  in  cubic  feet,   equation 
(84),  in  which  ;>  is  pounds  per  square  foot. 
tr,  Weight  of  a  cubic  foot  of  steam  =  -  • 

//,  The  heat  in  one  pound  of  liquid  above  32°  F.  in  thermal  units, 
equation  (9.~>),  page  111.  The  values,  however,  are  the  direct  results  of 
Kegnault's  experiments. 

Ae.  The  latent  heat  of  evaporation  in  thermal  units,  equation  (10), 
page  409. 

h,  The  total  heat  of  steam  above  32°  F.  l>eing  equal  to  h  -f  ?ie. 

SATURATED    STEAM. 


P- 

T. 

V. 

w.                 h. 

Ae. 

h. 

0.089 

32.0 

3b87. 

000029           0.0 

1091.7 

1091.7 

0.2 

54.0 

1482. 

0.00066          22.1 

1076.3 

1098.4 

1 

102.0 

335. 

0.00299          70.0 

1043.0 

1113.1 

5 

162.3 

73. 

0.01366        130.7 

1000.8        1131.5 

10 

193.2 

38. 

O.OM31 

161.9 

979.0        1140:9 

15 

213.0 

26.2 

0.03826 

181.8 

965.1      i  1146.9 

20 

228.0 

19.9 

0.05023        196.9 

954.6     1  1151  5 

25 

240.0 

16.1 

0.06199 

209.1 

946.0     j  1155.1 

30 

250.3 

13.6 

0.07360 

219.4 

938.9 

1158.3 

35 

259.2 

11.8 

0.08508 

228.4 

932.6 

1161.0 

40 

267.1 

10.4 

0.09644 

236.4 

927.0 

1163.4 

45 

274.3 

9.3 

0.1077 

243.6 

922.0 

1165.6 

50 

280.9 

8.4 

0.1188 

250.2 

917.4 

1167.6 

55 

286.9 

7.7 

0.1299 

256.3 

913.1 

1169.4 

60 

292.5 

7.1 

0.1409 

261.9 

909.3 

1171.2 

65 

297.8 

6.6 

0.1519 

267.2 

905.5 

1172.7 

70 

302.7 

6.15 

0.1628 

272.2 

902.1 

1174.3 

75 

307.4 

5.70 

0.1736 

276.9 

898.8 

1175.7 

80 

310.9 

5.49 

0.1843 

281.4 

895.6 

1177.0 

85 

315.2 

5.18 

0.1951 

285.8 

892.5 

1178.3 

90 

320.0 

4.858 

0.2058 

290.0 

889.6 

1179.6 

95 

323.9 

4.619 

0.2165 

294.0 

886.7 

1180.7 

100 

327.6 

4.403 

0.2271 

297.9 

884.0 

1181.9 

105 

331.1 

4.206 

0.2378 

301.6 

881.3 

1182.9 

110 

334.56 

4.026 

0.2488 

305.2 

878.8 

1184.0 

115 

337.86 

3.862 

0.2589 

308.7 

876.3 

1185.0 

120 

341.05 

3.711 

<r_>i;'.i.-, 

312.0 

S74.0 

11S«.0 

SATUKATED   STEAM. 

SATURATED   STEALS— Continued. 


465 


p- 

r. 

V. 

w. 

h. 

he. 

feu 

125 
130 
135 

344.13 
347.12 
350.03 

3.572 
3.444 
3.323 

0.2800 
0.2904 
0.3009 

315.2 
318.4 
321.4 

871.7 
869.4 
867.3 

1186.9 

1187.8 
1188.7 

140 
145 
150 

352.85 
355.59 
358.26 

3.212 
3.107 
3.011 

0.3113 
0.3218 
0.3321 

324.4 
327.2 
330.0 

865.1 
863.2 
861.0 

1189.5 
1190.4 
1191.2 

155 
160 
165 

360.86 
363.40 

365.88 

2.919 

2.833 
2.751 

0.3426 
0.3530 
0.3635 

332.7 
335.4 
338.0 

8593 

857.4 
855.6 

1192.0 
1192.8 
1193.6 

170 
175 
180 

368.29 
870.65 
372.97 

2676 
2.603 
2.535 

0.3737 
0.3841 
0.3945 

340.5 
343.0 
345.4 

853.8 
852.0 
850.3 

1194.3 
11950 
1195.7 

185 
190 
195 

375.23 
377.44 
379.61 

2.470 

2.408 
2.349 

0.4049 
0.4153 
0.4257 

347.8 
350.1 
352.4 

848.6 
847.0 
845.3 

1196.4 
1197.1 
1197.7 

200 
205 
210 

381.73 

383.82 
385.87 

2.294 
2.241 
2.190 

0.4359 
0.4461 
0.4565 

354.6 
356.8 
358.9 

843.8 
842.2 
840.7 

1198.4 
1199.0 
1199.6 

215 
220 
225 

387.88 
389.84 
391.79 

2.142 
2.096 
2.051 

0.4669 
0.4772 
0.4876 

361.0 
363.0 
365.1 

839.2 
837.8 
836.3 

1200.2 
1200.8 
1201.4 

230 
235 
240 

393.69 
395.56 
397.41 

2.009 
1.968 
1.928 

0.4979 
0.5082 
0.5186 

367.1 
369.0 
371.0 

834.9 
833.6 
832.2 

120.2.0 
1202.6 
1203.2 

2-15 
250 
255 

399.21 
400.99 
402.74 

1.891 
1.854 
1.819 

0.5289 
0.5393 
0.5496 

372.8 
374.4 
376.5 

830.9 
829.5 

8283 

1203.7 
1204.2 
1204.8 

260 
265 
270 

404.47 
406.17 
40785 

1.785 
1.753 
1.722 

0.5601 
0.5705 
0.5809 

378.4 
380.2 
381.9 

826.9 
825.6 
824.4 

1205.3 
1205.8 
1206.3 

275 

280 
285 

409.50 
411.12 
412.72 

1.691 
1.662 
1.634 

0.5913 
0.6020 
0.612 

383.6 
385.3 
387.0 

823.2 
821.8 
820.6 

1206.8 
1207.1 
12076 

290 
295 
300 

414.32 

415.87 
417.42 

1.607 
1.580 
1.554 

0.622 
0.633 
0.644 

-  388.6 
390.3 
391.9 

819.5 
818.3 
817.2 

1208.1 
1208.6 
1209.1 

305 
310 
315 

418.92 
420.42 
421.92 

1.529 
1.505 
1.481 

0.654 
0.664 
0.675 

393.5 
395.0 
396.6 

816.1 
815.1 
814.0 

1209.5 
1210.1 
1210.6 

320 
325 
330 

335 

42337 
424.82 
426.24 

'  427.64 

1.459 
1.437 
1.415 

1.395 

0.685 
0.696 
0.707 

0.717 

398.1 
399.6 
401.1 

402.6 

813.0 
811.7 
810.6 

809.5 

1211.1 
1211.3 
1211.7 

1212.1 

466 


SATURATED    AMMONIA. 


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CO    CO          CO    CO 
IO    O         IO    o' 


O    W         0100         10010         0100         100    10         0100         >00 
lOiO         CO5D£-         C-0000         OSOSO         OTHTH         <MC*CO         CO"^1 


INDEX. 


PAGE 

Absolute  scale 9,  40,401 

"       temperature 9, 116,  398 

zero 8,10,116,401,419 

Absorption  system 353 

Actual  heat  energy 4,  23 

Adiabatic  compression . .  185 

"        definition  of 16 

"        equations  for  ammonia  gas..  334 
"  "    imperfect'gases  148 

"  "   perfect        "       61 

"  "    saturated  steam, 

153,  177,  192,  426 

"          "   saturated  vapor  184 

"  "    superheated  steam, 

151 

"         expansion  of  saturated  steam, 

175, 177, 180,  192,  206 

lines 16 

Air,  a  perfect  gas 9 

"    an  imperfect  gas 13,  395 

"    compression  of 404 

"    compressor 63,  301 

"    elasticity  of 74,437 

"    engine 16,  169 

"    expansion  of 8,  114 

"    friction  of,  in  pipes 306 

"    required  for  combustion 262 

"    specific  heat  of 53,  382 

"    thermometer 7, 116,  395 

Ammonia 325-356 

"         engines 274 

"        gas,  density  of 327 

"  "    equations  of 330 

' '    specific  heat  of 326,  463 

"  "    volume  of  pound..  ..327,  333 

"        latent  heat  of 330 

"  "         "    "  evaporation... 328, 

332,348 

liquid,  density  of 326 

specific  heat  of .  335,  457 

"       volume  of  pound. . .  333 

Vi  pressure  of 325 


PAGE 

Ammonia,  saturated,  table  of 466 

' '  some  properties  of 325-333 

"  superheated 344 

"  tests  of 348,450 

"          vapor,  adiabatics  of. 334 

"  "        isothermals  of 333 

"  "       saturated,     specific 

heat,  a35, 456 

"  "       specific  volume 333 

Ash 363 

Atmosphere,  pressure  of 7 

"  height  of  homogeneous. .     76 

Atomic  weight 359 

Binary  vapor  engine 278 

Boiling  point 94,104,435 

Brine  325 

British  thermal  unit  (B.  T.  U.) 3,  24 

Caloric 3 

Carbonic  acid  gas,  equation  of 13 

Carnot's  cycle 17,20,36 

"        function 42 

Centigrade  scale 7 

Change  of  state 88 

Chemical  equivalent 359 

Chimney,  height  of 366 

Circulating  fluid 324 

Clearance 197 

Coal,  heat  of  combustion  of 221 

"     per  horse-power  perhour.181, 183, 221 

Combustion 358 

air  required  for  262.364 

heat  of 181,220,360,434 

Compound  engine 210, 222 

Compressor,  air 301 

workof 337 

Condensation  212 

Condensed  steam  due  to  expansion —  178 

Condenser 19,154 

work  of  *37 

Critical  temperature 104,  418 

Curve  of  saturation 100.  104 

Cut-off,  economical  200.  433 


472 


INDEX. 


PAGE 

Cycle,  definition  of 17 

Carnot's 17,36,41,  183 

"       non-revereible 20 

reversible 20 

Diagram,  ideal  of  steam 171 

"        indicator 20 

Differential,  exact 134, 146,446 

Disgregation,  heat  of 410,412 

Draft,  forced 365 

Duty  of  injector 285 

"      "pump  183 

"     "  refrigerating  plant 340 

Efficiency,  definition 159 

"          effect  of  superheating  on...  179 
"              "      "  specific  heat   and 
latent  heat  of  evapora- 
tion on 193,275 

••  ideal 219 

"          maximum,  formula  for 276 

"          of  boiler 111,221 

"  "  compressor 305 

"  '•  elementary  engine 33,163 

"  "  engine.  159,  193,  221,  228,  289, 

275 

"          "  injector 284 

44        •  "  naphtha  engine 270 

"  plant..  .159,  183,  220,  287,  447 
"  refrigeration.321,338,349,351 

"          "  regenerator  168 

"  "  steam  .  175-178,  182.  193,  200 

11  "  vapor  engines 427 

"          theoretical  method  of  im- 
proving   163 

Electricity  a  form  of  energy 2 

Energy  1,3,28,27 

"       effective  against  piston 173 

"       expended,  mechanical 167 

intrinsic 129 

"       measurement  of 1 

Engine,  air 169,  225,  226,  296 

"         ammonia 274 

"         Binary 278 

"         compressed  air 296 

Ericsson's  hot-air 234 

ether  424 

gas 247,248,253 

"         heat 17,159,  169 

"         multiple  expansion 210 

naphtha  266,270,271 

perfect  elementary.  18,  19,  159,  161 

"         steam  169-184 

"         Stirling's  hot-air ..221,284 

vapor 184.424,427 


PAGB 

Entropy 136-143 

Equation,  general,  of  fluids. .  .48,  126.  131, 
1:37.  4-M 

"          ofgas 10,13,330 

"          represented  geometrically,  428, 
442 

Ericsson's  hot-air  engine  234-246 

Escaping  gas,  weight  of 83 

Etherengine 424 

"      luminiferous 369-387 

"      specific  heat  of 424 

"      temperature  of  . .  379 

Evaporation,  factor  of 112 

latent  heat  of 9»,  11 1,  406. 

412 
"  "       "  apparent  and  real  112 

"  total  heat  of 110 

without  boiling 106 

Evaporative  power Ill 

Exact  differential 124,  146,  446 

Expansion,  adiabatic . .  15,  61,  175,  ITT.  I'.c,'' 
906,  253.  297 

isothermal.. 14,  54,  172,  205,  298. 
398 

latent  heat  of 5.38,86,127 

of  gas,  free 114 

"  ratio  of 157, 172,  197.  £45 

Expansive  force  of  congealing  water. .    HO 
Experiments,  Clement  and  Desormes'.    79 

Emery's 211 

Fairbairn  and  Tate's. ...  104 
Gately  and  Kletzsch ....  215 

Hirn's 213 

Isherwood's 214 

Joule's 24,114 

.    Major  Williams' 90 

Mayer's 78 

Mousson's 92 

Regnault's 7,  18,79 

"  Rowland's 25 

"  Sir  Win.  Thomson's.. 01,  114 

"  Zeuner's 151 

External  work.. 23,  55,  56,  85,  118,  160,  390, 
419 

Factor  of  evaporation 112 

FahrenheiO-cale 7 

Fire,  temperature  of      865 

Flow  of  gases 81 

"     "  steam 449 

Fluid,  circulating 824 

imperfect  12.85-158 

perfect 8,40,49-84 

Forced  draught 365 


INDEX. 


473 


PAGE 

Freezing  point  of  water 88,  91,  405 

"      "alcohol 405 

Friction  of  air  in  pipes 306 

Fuels,  heat  of  combustion  of 434 

Function,  Carnot's 42 

theroodynamic ..136,447 

Fundamental  equations 48 

Furnace 19,  256 

Fusion,  latent  heat  of 88 

Gas  analysis 261,  360 

"    boiling  points  of 435 

"    engine 247,437 

"    equation  of 10 

"    flow  of  81 

"    free  expansion  of 114 

"    heat  of  combustion  of 261 

*'    imperfect 12,  395 

"    perfect 8,40,49-84 

Gases,  table  of  properties  of 463 

General  formulas 48,  126,  131,  442 

Geometrical  representation  of  formula?, 

428,442 

Heat  absorbed 5,  22, 138, 145,  193 

"  "      general  equation  for..  .49, 85 

"  "      path  arbitrary 57,85 

"           "      pressure  and  volume  con- 
stant   107 

"  "      pressure  constant. 56, 120, 131 

"          "      temperature  constant . .  54, 86, 

131 

"          "      volume  constant.. 55, 119,  131 

"    energy 1,  2 

"    actual 4 

"    engine 17,159,317 

"    expended  per  pound  steam.  .173,  176, 
177 

"  "  "  cubic  foot  steam...  176 

"    in  liquid  and  vapor  combined ....  143 

"    internal 4,  410 

"    latent 5,  94,  328,  411 

"        "    of  evaporation 94,111,410 

"        "      "expansion 5,38,86,127 

"      "  fusion 88 

"    measurement  of 2 

"    mechanical  equivalent  of 24,  77 

"    nature  of 1 

"    of  combustion.  .181,  220,  26!4  360,  434 

"      "  disgregation 410 

"      "the  sun 369 

"    total  of  steam 110,  407 

"        "    "  several  vapors 407 

"    transmitted  through  plates 437 

"    velocity....  2 


Heights,  measurement  of 106 

Horse-power  per  pound  of  steam 180 

Hot-air  engine,  Ericsson's 234-246 

"       Stirling's 2231234 

Ice  formed  per  hour  per  horse-power. .  324 

"    latent  heat  of  fusion  of 89 

"    making  plant. . .   453 

"   melting  point  of 91 

"   specific  gravity  of 436 

Imperfect  fluids 12,  85-158 

gas 12,395 

Indicator  diagram  20 


from  compressor 350 

Injector 279 

"      compared  with  pump 289 

"      theory  of 280 

Internal  work  .4,  23, 85,  86, 119,  128,  390, 419 

Intrinsic  energy 120 

Inversion,  temperature  of 186,  425 

Isengeric  lines  397 

Isentropic    "  16,  39T 

Isobar  "  39?' 

Isodiabatic  "    157 

Isodynamic" 397 

Isometric     "  397 

Isopiestic     "  397 

Isothermal  expansion  .  14,  54,  172,  205,  298, 


"       lines 14, 

Joule's  equivalent 24 

Latent  heat 5,  94,  328,  411 

"       "    apparent  and  real 418 

"        "    of  evaporation  94,  111,  406,  412 

"      "expansion 5,38,86,127 

"       "     "  fusion 88 

Law,  first  of  thermodynamics 27 

tl    ofGay-Lussac 12 

"     "  Mariotte 12 

"    second  of  thermodynamics... 32, 134, 


Light,  mechanical  equivalent  of 436 

"      is  transmitted 2,369 

Lines,  adiabatic 16 

"       isentropic 16 

"       isothermal 14 

"       thermal 13,  397 

Liquefaction,  effect  of  pressure  on. ...  419 

"  of  steam  in  cylinder 214 

Liquefied  gases,  density  of 436 

Liquid  and  its  vapor  combined 143 

"       "    solids,  table  of  properties 

of  469 

"       isothermal  of  a 103 


4T4 


INDEX. 


Logarithmic  tables 458 

Luminous  ether 369,387 

"    rarityof 375 

Marine  engines 216-219 

Mariotte's  law 18 

Matter,  incombustible 263 

Mechanical  equivalent  of  heat 24,  77 

"          mixtures 108 

Melting  point 88,91,92,  405 

Miscellaneous 216 

Multiple  expansions 210 

Naphtha  engine 266 

Otto  gas  engine 250 

Oxygen,  curve  of  saturation  of 435 

Path  of  fluid 14,57,85 

Perfect  gas 8,40,49-S4 

Practical  considerations 195,  208 

Pressure  and  temperature  constant 97 

"         "    volume  constant 107 

"       constant 29 

"       equivalent  to  heat  expended.  175 

mean  effective 175, 178, 198 

"       mean  total  forward 175 

"       of  steam  of  marine  engines.  219 

Priming 420 

Pulsometer 2:12 

Quadruple  expansion 210,  220 

Ratio  of  expansion 154,  172,  197,  245 

"      "          "        with  clearance....  197 

Reduction  table 461 

Refrigerating  machine 818,  450 

"  system,  vapor  saturated.  387 

Refrigerator 19,161,  223 

Regenerator 166,223 

Relative  xpeciflc  beat*  68 

Reversible  engine 397 

Saturated  steam,  adiabatic  for...  158,  184 

densityof.. 104 

tableof 464 

"  "       formulae  for 97 

"  "       volume  of 97 

"         vapor,  specific  heat  of .  .145,  428 

Saturation,  curve  of 100,  1O4,  435 

Scale,  absolute 9,40,  401 

"       Thomson's 42,398,420 

Solids,  specific  heat  of,  tables 108,  462 

Sound,  velocity  of 75,398 

Source  19 

Space,  heat  of 369 

Specific  gravity  of  ice 436 

"      heat. . .  .29,  49,  58,  77, 117,  120,  407 

"       "    apparent 120 

"       "    at  change  of  state    117 


PAGE 

Specific  heat  at  constant  pressure.. 29,  122, 
264 

"       "     "       "         volume.... 31, 53, 
264 

"       "    constant 49 

"    difference  of 49,  122,  440 

"       "    dynamic 120,  145 

"       "    expression  for.  .30,  31,  117-131 
"       "    mean,  expression  for  ....  126 

•'       "    of  air 53,  382 

"        "     "  ammonia  gas 326 

••        "     "         "         liquid.... 337, 456 

"       "     "  any  substance 117 

"       "     "  ether 424 

"       "     "  luminiferous  ether. ...  883 
it       it     •«  saturated  vapor...  145,  423 

"       "     "  solids 108 

"        "     "  water 90,  126,  264 

"       "    ratio  of  (y). . .  54,  75,  264,  404 
'•       "       "     ""  methods  for  de- 
termining     79 

"    real 120 

"    relative 63 

"     volume 10,9J;383 

"     ofair 10 

"          "      "  ammonia  gas 338 

State,  change  of 88 

Steam,  condensed,  due  to  expansion. .  178 
"     efficiency  of .  ..175-178,  ?82,  198,  200 

"     engine 169,213 

"     gas 118 

"     Injector 279 

"     jacket 195 

"     liquefaction  of 148 

"     per  hour  per  horse-power.  .180,  216 

"     practice  437 

"     pomp 181 

"     relation  of  pressure  to  boiling 

pointof 94 

"     saturated 96 

"  "       densityof 104 

"  "        expansion  of .  177, 184,  426 

"  "       formula:  for,  97,  411-414. 

417 

specific  heat  of...  145,  420 

tableof 464 

total  heat  of  .   ...110,  407 

"     ship  records  218.  436 

"     specific  heat  of 463 

"     superheated 113,  417,  418 

expansion  of 172 

"     to  find  y  for 150 

"     turbine ...  308 


INDEX. 


475 


PAGE 

Steam,  volume  of  pound  of 98 

Stellar  heat 379 

Stirling's  hot-air  engine 223-234 

Sublimation 106 

Sulphur  dioxide 357,  436,  449,  468 

Sun,  heat  of 369 

Superheated  steam 113 

Superheating 195,  344,  421 

Tables,  gases,  properties  of 463 

"     logarithmic 458 

"      liquids  and  solid  properties  of  462 

"      reduction 461 

"     saturated  ammonia 466 

steam 464 

"     sulphur  dioxide 468 

Temperature 5 

absolute 9,  116,  398 

"         critical 104,418 

"    '    final,  how  to  find 61,298 

of  fire 365 

"  "  inversion 186,  425 

"  space 436 

to  change  from  F.  to  C....  406 

Terminal  pressures 175 

Test  of  absorption  plant 355 

"    "  naphtha  engine 267 

"    "  Otto  gas-engine 259 

"    "  pulsometer 452 

"    "  refrigerating  plant.. 348,   352,  448 
450 

"    "  steamships 218-222 

Thermal  capacity 28 

lines  J3,  397 

"        unit 3 

Thermodynamic  function 136,  447 

"  surface 10 

Thermodynamics,  first  law 27 

principles  of..  ..161,  275 

second  law.. 32,  389-393, 

397 

Thermometer 6,7 

;'  air 7 

"  "  zero  of 8 

Triple  expansion  engines 210,  228 

Turbine,  steam 308 

Unit  of  evaporative  power Ill 

Vacuum,  extent  of  375 


PAGE 

Vapor 96 

"    combined  with  liquid 143 

"    densities  compared  with  air —    412 

"    engines 184,424,427 

"       Binary 278 

"    ofammonia  333 

"    relation  between  pressure   and 

"      temperature  of 97 

"  specific  heat  of  saturated. .  ..145,  423 

"    totalheatof 94,407 

"    volume  of 98 

"   weight  of 99 

Velocity  of  gas 82 

"       "  gravitation 378 


2 

"  light 2,  369 

"        "  sound 75,  373 

"       "  steam  through  pipe 195 

"  water       "       "    281 

"       "      "      from  orifice 283 

"        "  wave 71-75 

Virtual  pressure .  85 

Volume  of  vapor 98 

"      specific 10 

Water  consumed  per  I.  H.  P 182 

"     expansion  of 102 

"     maximum  density  of 3 

"     volume  of  one_ponnd  of 98 

"     weight  of 102,281 

Watt's  law  436 

Wave,  velocity  of 71  -75 

Wiredrawing 195 

Work 3 

"    done 19,390 

"        "    by  compression 193 

"    "   injector 284 

"       "    per  en.  ft.  steam 17S 

"    "  pound      "     ....175,  177,  180 

"    entire,  due  to  expansion 85,  175 

"    external.23,  55,  56, 85, 118, 160,390,419 

"  "       during  evaporation 410 

"  "  increment  of,  equal  to 
increment  of  heat  ab- 
sorbed  403 

"       internal.  .4,  23,  85,  86,  119,  103,  128, 


Zero,  absolute 8, 10,  116,  401,  -109 


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Bridge  Trusses 8vo,  250 

Roof  Trusses 8vo,  125 

Howe's  Treatise  on  Arches Svo,  4  00 

Johnson's  Modern  Framed  Structures Small  4to,  10  00 

Merriman  &  Jacoby's  Text-book  of  Roofs  and  Bridges. 

Part  L,  Stresses 8vo,  250 

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Part  II.,  Graphic  Statics 8vo,  2  50 

Merriman  &  Jacoby's  Text-book  of  Roofs  and  Bridges. 

Part  III.,  Bridge  Design Svo,  2  50 

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*  Morison's  The  Memphis  Bridge Oblong  4to,  10  00 

Waddell's  Iron  Highway  Bridges Svo,  4  00 

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16mo,  morocco,  3  00 

Wood's  Construction  of  Bridges  and  Roofs Svo,  2  00 

Wright's  Designing  of  Draw  Spans.     Parts  I.  and  II..8vo,  each  2  50 

"              "          "      "          "         Complete Svo,  3  oO 

4 


CHEMISTRY— BIOLOGY-PHARMACY. 

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Adriauce's  Laboratory  Calculations 12mo,  f  1  25 

Allen's  Tables  for  Iron  Analysis 8vo,  3  00 

Austen's  Notes  for  Chemical  Students 12mo,  1  50 

Bolton's  Student's  Guide  in  Quantitative  Analysis 8vo,  1  50 

Boltwood's  Elementary  Electro  Chemistry (In  the  press.) 

Classen's  Analysis  by  Electrolysis.  (HerrickandBoltwood.).8vo,  3  00 

Cohu's  Indicators  and  Test-papers 12mo  2  00 

Crafts's  Qualitative  Analysis.     (Schaeffer.) 12mo,  1  50 

Davenport's  Statistical  Methods  with  Special  Reference  to  Bio- 
logical Variations 12rno,  morocco,  1  25 

Drechsel's  Chemical  Eeactions.    (Merrill.) 12mo,  1  25 

Fresenius's  Quantitative  Chemical  Analysis.    (Allen.) 8vo,  6  00 

"         Qualitative          "              "            (Johnson.) 8vo,  300 

(Wells.)         Trans. 

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Fuertes's  Water  and  Public  Health.. 12mo,  1  50 

Gill's  Gas  and  Fuel  Analysis 12mo,  1  25 

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Helm's  Principles  of  Mathematical  Chemistry.    (Morgan).  12mo,  1  50 

Ladd's  Quantitative  Chemical  Analysis 12mo,  1  00 

Landauer's  Spectrum  Analysis.     (Tingle.) 8vo,  3  00 

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Mason's  Water-supply , 8vo,  5  00 

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Miller's  Chemical  Physics 8vo,  2  00 

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Morgan's  The  Theory  of  Solutions  and  its  Results 12mo,  1  00 

Elements  of  Physical  Chemistry ; 12mo,  200 

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O'Brine's  Laboratory  Guide  to  Chemical  Analysis 8vo,  2  00 

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5 


Poole's  Calorific  Power  of  Fuels 8vo,  $3  00 

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Schimpf's  Volumetric  Analysis 12mo,  2  50 

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Stockbridge's  Rocks  and  Soils Svo,  2  50 

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Whipple's  Microscopy  of  Drinking-water Svo,  3  50 

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Wulling's  Inorganic  Phar.  and  Med.  Chemistry 12mo,  2  Oft 

DRAWING. 

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MacCord's  Descriptive  Geometry Svo,  3  00 

"          Kinematics Svo,  500 

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Reid's  A  Course  in  Mechanical  Drawing Svo.  2  00 

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6 


Warren's  Plane  Problems 12mo,  $1  25 

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Warren's  Shades  and  Shadows 8vo,  3  00 

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Holman's  Precision  of  Measurements 8vo,  2  00 

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*Tillman's  Heat 8vo,  1  50 

7 


ENGINEERING. 

CIVIL — MECHANICAL— SANITARY ,  ETC. 

(See  also  BUIDGES,  p.  4 ;  HYDRAULICS,  p.  9 ;  MATERIALS  OF  EN- 
GINEERING, p.  10;  MECHANICS  AND  MACHINERY,  p.  12  ;  STEAM 
ENGINES  AND  BOILERS,  p.  14.) 

Baker's  Masonry  Construction Svo,  $5  00 

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Black's  U.  S.  Public  Works Oblong  4to,  5  00 

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Byrne's  Highway  Construction Svo,  5  00 

"       Inspection  of  Materials  and  Workmanship 16mo,  3  00 

Carpenter's  Experimental  Engineering  Svo,  6  00 

Church's  Mechanics  of  Engineering— Solids  and  Fluids 8vo,  G  00 

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Crandall's  Earthwork  Tables Svo,  1  50 

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Howe's  Retaining  Walls  (New  Edition.) .12mo,  1  25 

Hudson's  Excavation  Tables.     Vol.  II Svo,  1  00 

Hutton's  Mechanical  Engineering  of  Power  Plants Svo,  5  00, 

"         Heat  and  Heat  Engines Svo,  500 

Johnson's  Materials  of  Construction Large  Svo,  6  00 

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Kent's  Mechanical  Engineer's  Pocket-book 16mo,  morocco,  5  00 

Kiersted's  Sewage  Disposal 12mo,  1  25 

Mahau's  Civil  Engineering.      (Wood.) Svo,  5  00 

Merriman  and  Brook's  Handbook  for  Surveyors. . .  .16mo,  mor.,  2  00 

Merriman's  Precise  Surveying  and  Geodesy Svo,  2  50 

Retaining  Walls  and  Masonry  Dams Svo,  2  00 

Sanitary  Engineering Svo,  200 

Nagle's  Manual  for  Railroad  Engineers 16mo,  morocco,  3  00 

Ogdeu's  Sewer  Design 12mo,  2  00 

Pattou's  Civil  Engineering Svo,  half  morocco,  7  50 

8 


Pulton's  Foundations 8vo,  $5  00 

Pratt  and  Alden's  Street-railway  Road-beds 8vo,  2  00 

Rockwell's  Roads  and  Pavements  in  France 12tao,  1  25 

Searles's  Field  Engineering  . . . . : 16mo,  morocco,  3  00 

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Siebert  and  Biggin's  Modern  Stone  Cutting  and  Masonry. .  .8vo,  1  50 

Smart's  Engineering  Laboratory  Practice 12mo,  2  50 

Smith's  Wire  Manufacture  and  Uses Small  4to,  3  00 

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Taylor's  Prisuioidal  Formulas  and  Earthwork 8vo,  1  50 

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*  "          Cross-section Sheet,  25 

Excavations  and  Embankments Svo,  2  00 

Laying  Out  Curves 12mo,  morocco,  2  50 

Waddell's  De  Pontibus  (A  Pocket-book  for  Bridge  Engineers). 

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Wait's  Engineering  and  Architectural  Jurisprudence Svo,  6  00 

Sheep,  6  50 

"      Law  of  Field  Operation  in  Engineering,  etc Svo. 

Warren's  Stereotomy — Stone-cutting Svo,  2  50 

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Wegmanu's  Construction  of  Masonry  Dams 4to,  5  00 

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Wheeler's  Civil  Engineering Svo,  4  00 

Wolff's  Windmill  as  a  Prime  Mover Svo,  3  00 


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Bazin's  Experiments  upon  the  Contraction  of  the  Liquid  Vein. 

(Trautwine. ) 8vo,  2  00 

Bovey 's  Treatise  on  Hydraulics Svo,  4  00 

Coffin's  Graphical  Solution  of  Hydraulic  Problems 12mo,  2  50 

Ferrel's  Treatise  on  the  Winds,  Cyclones,  and  Tornadoes. .  .8vo,  4  00 

Fol well's  Water  Supply  Engineering Svo,  4  00 

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Herschel's  115  Experiments  Svo,  2  00 

9 


Kiersted's  Sewage  Disposal 12mo,  $1  25 

Mason's  Water  Supply 8vo,  5  00 

"    Examination  of  Water 12mo,  1  25 

Merriman's  Treatise  on  Hydraulics 8vo,  4  00 

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Weisbach's  Hydraulics.     (Du  Bois.) 8vo,  500 

Whipple's  Microscopy  of  Drinking  Water 8vo,  3  50 

Wilson's  Irrigation  Engineering 8vo,  4  00 

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Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  00 

Wood's  Theory  of  Turbines 8vo,  2  50> 

MANUFACTURES. 

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Allen's  Tables  for  Iron  Analysis 8vo,  3  00 

Beaumont's  Woollen  and  Worsted  Manufacture 12mo,  1  50 

Bolland's  Encyclopaedia  of  Founding  Terms 12mo,  3  00 

"         The  Iron  Founder 12mo,  250 

"          "       "          "        Supplement 12mo,  250 

Bouvier's  Handbook  on  Oil  Painting .  ..12mo,  2  00 

Eissler's  Explosives,  Nitroglycerine  and  Dynamite 8vo,  4  00 

Ford's  Boiler  Making  for  Boiler  Makers 18uio,  1  00 

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*  Reisig's  Guide  to  Piece  Dyeing 8vo,  25  00 

Spencer's  Sugar  Manufacturer's  Handbook 16mo,  morocco,  2  00 

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Thurston's  Manual  of  Steam  Boilers 8vo,  5  00 

Walke's  Lectures  on  Explosives 8vo.  4  00 

West's  American  Foundry  Practice 12mo,  2  50 

"      Moulder's  Text-book  12mo,  2  50 

Wiechmann's  Sugar  Analysis Small  8vo,  2  50 

Woodbury's  Fire  Protection  of  Mills 8vo,  2  50 

MATERIALS  OF  ENGINEERING. 

STRENGTH— ELASTICITY—  RESISTANCE,  ETC. 
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Baker's  Masonry  Construction 8vo,  5  00 

Beardslee  and  Kent's  Strength  of  Wrought  Iron 8vo,  1  50 

Bovey's  Strength  of  Materials 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  Materials 8vo,  5  00 

10 


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Church's  Mechanics  of  Engineering— Solids  and  Fluids 8vo,  6  00 

Du  Bois's  Stresses  in  Framed  Structures Small  4to,  10  00 

Johnson's  Materials  of  Construction 8vo,  G  00 

Lanza's  Applied  Mechanics 3vo,  7  50 

(  Marteus's  Testing  Materials.     (Henuing.) 2  vols.,  8vo[  7  50 

'  Merrill's  Stones  for  Building  and  Decoration 8vo,  5  00 

Merriman's  Mechanics  of  Materials 8vo,  4  00 

"          Strength  of  Materials 12mo,  100 

Pattou's  Treatise  on  Foundations 8vo,  5  00 

Rockwell's  Roads  and  Pavements  in  France 12mo,  1  25 

Spaldiug's  Roads  and  Pavements 12mo,  2  00 

Thurstou's  Materials  of  Construction 8vo,  5  00 

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Vol.  I,  Non-metallic 8vo,  200 

Vol.  II.,  Iron  and  Steel 8vo,  3  50 

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Baker's  Elliptic  Functions 8vo,  1  50 

Barnard's  Pyramid  Problem 8vo,  1  50 

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Briggs's  Plane  Analytical  Geometry 12mo,  1  00 

Chapman's  Theory  of  Equations 12mo,  1  50 

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Davis's  Introduction  to  the  Logic  of  Algebra 8vo,  1  50 

Halsted's  Elements  of  Geometry t..8vo,  1  75 

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Johnson's  Curve  Tracing 12mo,  1  00 

"        Differential  Equations — Ordinary  and  Partial. 

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(In  the  press. ) . 

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*      "        Trigonometry  with  Tables.     (Bass.) 8vo,  300 

*Mahan's  Descriptive  Geometry  (Stone  Cutting) 8vo,  1  50 

Merriman  and  Woodward's  Higher  Mathematics 8vo,  5  00 

Merriman's  Method  of  Least  Squares 8vo,  2  00 

Rice  and  Johnson's  Differential  and  Integral  Calculus, 

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Totteu's  Metrology 8vo,  2  50 

Warreu's  Descriptive  Geometry 2  vols.,  8vo,  3  50 

Drafting  Instruments 12mo,  125 

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Linear  Perspective 12mo,  100 

Primary  Geometry 12mo,  75 

Plane  Problems 12mo,  125 

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Baldwin's  Steam  Heating  for  Buildings 12mo,  2  50 

Barr's  Kinematics  of  Machinery Svo,  2  50 

Benjamin's  Wrinkles  and  Recipes 12mo,  2  00 

Chordal's  Letters  to  Mechanics 12mo,  2  00 

Church's  Mechanics  of  Engineering Svo,  6  00 

"        Notes  and  Examples  in  Mechanics Svo,  200 

Crehore's  Mechanics  of  the  Girder Svo,  5  00 

Cromwell's  Belts  and  Pulleys 12mo,  1  50 

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Compton's  First  Lessons  in  Metal  Working 12mo,  1  50 

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Vol.  III.,  Kinetics Svo,  3  50 

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Flather's  Dynamometers 12mo,  2  00 

Rope  Driving 12mo,  200 

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Johnson's  Theoretical  Mechanics.      An  Elementary   Treatise. 
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Jones's  Machine  Design.     Part  I.,  Kinematics 8vo,  1  50 

12 


Jones's  Machine  Design.     Part  II.,  Strength  and  Proportion  of 

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Lanza's  Applied  Mechanics 8vo,  7  50 

MacCord's  Kinematics. 8vo,  5  00 

Merriman's  Mechanics  of  Materials 8vo,  4  00 

Metcalfe's  Cost  of  Manufactures .8vo,  5  00 

•Michie's  Analytical  Mechanics 8vo,  4  00 

Richards's  Compressed  Air 12mo,  1  50 

Robinson's  Principles  of  t  Mechanism 8vo,  3  00 

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Thurston's  Friction  and  Lost  Work 8vo,  8  00 

The  Animal  as  a  Machine 12mo,  100 

Warren's  Machine  Construction 2  vols.,  8vo,  7  50 

Weisbach's  Hydraulics  and  Hydraulic  Motors.    (Du  Bois.)..8vo,  5  00 
Mechanics    of   Engineering.      Vol.    III.,    Part  I., 

Sec.  I.     (Klein.) 8vo,  500 

Weisbach's   Mechanics    of  Engineering.     Vol.   III.,    Part  I., 

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Weisbach's  Steam  Engines.     (Du  Bois.) 8vo,  500 

Wood's  Analytical  Mechanics 8vo,  8  00 

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"                                              Supplement  and  Key 12mo,  1  25 

METALLURGY. 

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Allen's  Tables  for  Iron  Analysis 8vo,  3  00 

Egleston's  Gold  and  Mercury Large  8vo,  7  50 

"         Metallurgy  of  Silver Large  8vo,  7  50 

*  Kerl's  Metallurgy— Copper  and  Iron 8vo,  15  00 

*  "           "               Steel,  Fuel,  etc 8vo,  1500 

Kunhardl's  Ore  Dressing  in  Europe 8vo,  1  50 

Metcalf's  Steel— A  Manual  for  Steel  Users 12mo,  2  00 

O'Driscoll's  Treatment  of  Gold  Ores 8vo,  2  00 

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Alloys 8vo,  250 

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Barringer's  Minerals  of  Commercial  Value. . .  .Oblong  morocco,  2  50 

Beard's  Ventilation  of  Mines 12mo,  2  50 

Boyd's  Resources  of  South  Western  Virginia 8vo,  3  00 

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Brush  and  Peufield's  Determinative  Mineralogy.   New  Ed.  8vo,  4  00 
13 


Chester's  Catalogue  of  Minerals 8vo,  $1  25 

"          "        "        Paper,  50 

Dictionary  of  the  Names  of  Minerals.. 8vo,  3  00 

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UNIVERSITY  OF  CALIFORNIA  LIBRARY 

Los  Angeles 
This  book  is  DUE  on  the  last  date  stamped  below. 


STf 


JAN  4     1961 

9RM    RM 

WARS     1961 
MAR  6 


• 


urn  1 


MAY  2  1 1962 
NAY      2  RBA 


8EF7 


lorm  L9-100m-9,'52(A3105)444 


Engineering 
Library 


A     000316625     3 


